Фундаментальная и прикладная математика (2001, №1)

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Abstract B. V. Novikov, Semigroup cohomologies: a survey, Fundamentalnaya i prikladnaya matematika, vol. 7 (2001), no. 1, pp. 1{18.

A survey of research in semigroup cohomologies and their applications is given.

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1] . . . .: , 1987. 84:1A352] 2] !" #. . $%"& // (). * +$. | 1964. | .. 33, / 2. | . 263{269. 65:1A207] 3] !" #. . $%"& " &23 // (). * +$. | 1965. | .. 39, / 1. | . 3{10. 65:11A225] 4] !" #. . $%"& " // (). * +$. | 1966. | .. 41, / 3. | . 513{520. 66:7A240] 5] !" #. . $%"& 44523 5"23 // (). * +$. | 1967. | .. 46, / 1. | . 11{18. 67:10A134] 6] !" #. . " 4 2" 5 2 6" 7 4 6" 7 "(2 3 ""& // .. .(. 4". -4 * +$. | 1975. | .. 48. 76:8A504] 7] 4 *., 97"(" . + 6"& "(. | .: :, 1960. 61:2A238] 8] 4 !" $. ;. < &3 5 // (). * +$. | 1985. | .. 117, / 3. | . 465{468. 86:1A492] 9] == *., >"4 +. * "(6"& 4" & . .. 1, 2. | .: , 1972. 64:10A191, 68:10A123] 10] >. ;. > 2 & // ?6. !. :+> . +"@". | 1971. | .. 404. | . 275{284. 71:12A163] 11] >. ;. . ;. + 6"& 6"& 34"4 ( ()" " 6"3 // !5. 5! 5. 4". | 1982. | / 5. | . 30{34. 82:9A340] 13] " .". < %" 5 // (). * +$. | 1976. | .. 83, / 1. | . 25{28. 77:5A279] 14] " .". < &3 5 // (). * +$. | 1977. | .. 85, / 3. | . 545{548. 77:12A424] 15] " .". < " 4 23 =4 23 5 7453 5 // (). * +$. | 1977. | .. 87, / 2. | . 281{284. 78:6A416] 16] 5 5 . B. < 0- &3 // ." . . 5 . ==. -7 "(. | "5: 5 , 1978. | . 185{188. 79:6A368] 17] 5 5 . B. < "4523 " 45"&3 // ; . * ?$, ". *. | 1979. | / 6. | . 474{478. 79:11A161] 18] 5 5 . B. 0- 5 " 0- 423 // B"4 #C.

. -4. | 1981. | B2. 46, / 221. | . 80{85. 82:6A356] 19] 5 5 . B. < 526" 7 " 4 23 // B"4 #C. . -4. | 1981. | B2. 46, / 221. | . 96. 82:6A357]

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20] 5 5 . B. 4" 7 4"!" 46" // .. .(. 4. -4 * +$. | 1982. | .. 70. | . 52{55. 83:5A341] 21] 5 5 . B. 6 & *. . & 5 I==@"4 5 &3 // (). * +$. | 1977. | .. 86, / 3. | . 546{548. 78:3A277] 32] >6 & *. . C2" %7" 52 %"& "7 // .. .(. 4". -4 * +$. | 1979. | .. 62. | . 76{90. 80:1A430] 33] >6 & *. . < !5 23 =4 3 =4 5 !6"& 5 4" "7 // .. .(. 4". -4 * +$. | 1986. | .. 83. | . 60{75. 87:4A437] 34] >6 & *. . < 3 // .. .(. 4". -4 * +$. | 1988. | .. 91. | . 36{43. 89:7A308] 35] J" . K., KC "7=" L. +. c !(x) = xp $ x 2 R+ 0 < p 6 19 8 x 0 6 x 6 2 > > >x ; 2 3 < x 6 4 : 2 x > 4: ', & & & ' . % ' R+ & , : 1) : 2). 2. A f g 2A | '( '( . 4 $ P!& $ c > 0 & !(r) = !c (r ), $ r8 = fr g 2 RA+. %' ( & . : 4) A RA+. . 0 $ " > 0 & U ' RA+, $ (80 8h) 2 U !(8h) < "2 , $ h8 2 RA+. : , U 6 ' jr0 ; r00 j < , $

> 0. (8r0 r800) 2 U . 7& r8 r = min(r0 r00), 2 A. 4 $ r8 6 r80 r800 ! ' 8h0 8h00 2 RA+, r8 + 8h0 = r80 , r8 + h8 00 = r800, 6 (8r 8h0 r80) (8r 8h00 r800) & $ . 7 , (80 8h0) (80 8h00) 2 U , !(8h0 ) !(8h00) < "2 . > : 3) !(8r ) ; !(8h0 ) 6 6 !(8r0 ) 6 !(8r) + !(8h0 ), !(8r ) ; !(8h00) 6 !(8r00 ) 6 !(8r) + !(8h00 ), j!(8r0) ; !(8r00 )j < ". : 5) A0 | A. 0 : RA+ ! R+A i0 : RA+ ! RA+

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24

. .

' $ : 3). , !S . U | RA+, !(8r ) < " r8 = fr g 2 U , $ " | 6 . : , U 6 r < , $ &$ A0 A, > 0 !& $ 2 A0 . 7& V ' X , (x1 x2) 2 V ( 2 A0 ! (x1 x2) < . 4 $, , (x1 x2) 2 V !S (x1 x2) < ". ( &( , : 3). (8r1 r82 r83) | RA+, &! $ . 2 A R2 $ P1 P2 P3 P1 P2 = r1 , P1 P3 = Qr2 , P2 P3 = r3 . X = R2 P1 = fP1 g, P2 = fP2 g, P3 = fP3 g | X . &' '( ( X R2 , 2 A, & . 4 $ (P1 P2) = = r1 , (P1 P3) = r2 , (P2 P3) = r3 ( 2 A,

S = f g, : !S (P1 P2) = !(8r1 ), !S (P1 P3) = !(8r2), !S (P2 P3) = !(8r3), &( : 3). L &( '( & $ . X = RA+. % R+ !, (0 r ) = r . A, '2, | & X S = f g, f (80 r8)g = r8 !S (80 r8) = !(8r ), ' & . 5 ' X , '( ' & . 5 1) B fA g2B f! g2B ! | , ) : 6), S = f g 2A , 2 B , | S X , A , S = S .

; Q ! ! S !, M = f(! )S g2B , %. .

S = f g 2A 2B &O $ & A ( X 9 S , A = A . M , ' B . . > :

Y

Y

! ! (x1 x2) = ! ! (f (x1 x2)g

S

2A 2B ) =

= !(f! (f (x1 x2)g 2A )g2B ) = !(f(! )S (x1 x2)g2B ) = ! (x1 x2) !&'( x1 x2 2 X .

25

5 2) * % % ! % S , ' , !S . . x1 x2 2 X x1 6= x2. 4 $ $

!S (x1 x2) = !(f (x1 x2)g) RA+

' . !S (x1 x2) 6= 0 : 9). 5 3) * % % ! , X !S , , S , $ A, !. . A0 | A, & , '( !, S 0 | S ,

A0 . 7& , '2, i0 RA+ RA+ !0 | $ !ji ( A+ ) . 4 $ $ , !S0 !S X !. > ' 1 , , !S , , 2, , S 0 . 0 & 2. 5 $ U ', S 0 . : , 6 < , $ &$ A00 A0 > 0. L 6 " > 0, $ ! f(x1 x2) 2 X X : !S (x1 x2) < "g U . 7& i R+ RA+ , $ 2 A00 . : & !0 ji( +) : 5) , : 8) " > 0, $ !0 ji( +) (x) < " " , (x1 x2) | , x < . " = 21 min 2A 0 0 X X , !S (x1 x2) = ! (f (x1 x2)g 2A ) < ". 7& r8 f (x1 x2)g 2A RA+ , r8 | $ ! ! i (R+ ), $ 2 A00. J $ $ ' ' !, ' , (x1 x2). 7 , r8 6 r8, !0 (8r ) 6 2!0(8r) < 2" 6 " (x1 x2) < A00. 5 4) &( ' ! . P &' $ , & S , ( ( ( S . 5 4) S1 S2 | X , S1 = S2 . % , %' ( , %' ) $ , . . 5 ' . 0 S = f g 2A $ !O $ & f : A ! S . %6 A 2 , 1 2 , f (1 ) = f (2 ), & A . 0

0

0

R

0

0

R

R

00

0

0

0

0

26

. .

A & & A ! S S ,

A . 0 , S S . $ , '& ! , A A, S | S . 4 $ & ! A & & ! A, . 5 !S = !S ', ' $ & S &' S . P &' & !, & Q $ a 2 A Q Qa $ & R+a R+ . 4 $ Qa 2a a2A '' (! Q Q RA+ RA+, '

R+ . 7& a2A 2a $ & D. 7$ D !& & ! A,

& ! A . 0 !&'( x1 x2 2 X ! f (x1 x2)g 2A 2 D, ', !S = !S . 3 & $ . 7& , S1 S2 ( & ! ' & . 5 5) | X , , S . ' %' !, , !S , % . . %'& S = f g 2A & 6

S 0 , &' $ ! S 0 . % 6 $ ( ! ! '( f"n g $ "n 6 '

& An A & T '( Qn = f g 2An , '( ! f(x1 x2) 2 X X : (x1 x2) < g

2An

f(x1 x2) 2 X X : (x1 x2) < "n g. 4 $ S 0 S 0

, A = An . n ! | ! & A, $ A0 . 5 3) ' . 4 & , , ! & 2, > 0, 2 A0 . 5 6) ' , , , ' %'

27

!, % X X

!S > .

. ' 5 1) !, $ S , & . @ , , 6 1. 5 ! R+, !

!^ (s) = sup (x1 x2)

(x1 x2 )6s

, &' !

& , . 7 , $ , !^ & ! : 1) 0 6 !^ (s) 6 1 !& $ s 2 R+9 2) !^ (0) = 09 3) !^ (s) 9 4) !^ (s) ' . % ! $ , , , , ! . 4 , ' ( !, $ $ ( ' ! & f(s t) 2 R2 : s 2 R+ 0 6 t 6 !^ (s)g ( ( ( $ '( ' '( ). % , ! ' ! $ , !& $ " > 0 k > 0, $ ' f(s t) 2 R2 : s 2 R+ 0 6 t 6 " + ksg, ! | !^ , & ! | 6 ' . 3 2 . % , !!$ $ ,

! : X Y | , 6 Y $ , f : X ! Y | ' &. 4 $ &

, X , $ & f ' . 7 , 5 6) &( . 6 6 $ $ . 7 $ RA+ ! ^(8r) = sup (x1 x2): f (x1 x2 )g6r

4 $ ! . 1. + !S , ) ^ % . 2. X | R | 2 . % '

28

. .

Y = X=R 6 ! ! (, '( ' & p : X ! Y . 7 & p ' $ ( & '() '( ( ''( & . Y $ $ , $ 6 & X . :' !, ( $ , !$ ( ! ! S , , ! ! Y . L 6 ' '( , $ $ 6 . A ,

' !& , . 6 ' (

& , & !& $ $ & , $ . * $ !( ( & . 5 4) , $ 6

. 5 !!, - & !, !. | X , y1 y2 | ' Y . R k , ! ' X , & ' !&! 2k-! Z = fx000 x01 x001 : : : x0k;1 x00k;1 x0kg X , !! ! : p(x000 ) = y1 , p(x0k ) = y2 , p(x0i) = p(x00i ) i = 1 : : : k ;1. 0 Z 6 Pk (Z ) = (x00i;1 x0i). 1 7 ^ Y , ^ (y1 y2 ) = Z(yinfy ) (Z ), 1 2 $ Z (y1 y2) & ( , !( y1 y2 . Y , ' & X , & ' ( (. 7 : 1) & & p Y , 9 2) ( , '( X ,

$ , $' 2 ' ( 9 3) X & . 5 $ S X & ' S^ Y , , ( ( S .

29

A Y ! , 6! ^ X , & p & '' . , Y 2, ( $ , 6'( S , . 3 !! ! Y & ' & , ! S . , S = f g 2A $ X ( , !( ( & ), ' ' Y !. , , | $ ( Y ) , | 6 & X . 4 $ S , ! . ( , S^, !! . 4 & , ! . 2. X | , R | , $ X S | X , %' . ) Y = X=R , %' S . K ' , $ X & , $ '( R. 7& F = ff g2B & X &. X 6 & $ , !&'( x1 x2 2 X , 2 B (f (x1) f (x2)) 6 (x1 x2). 5 ! X 6 & F , &' , &!( . , $ !$ $ & ! S = f g &'( F

& . % , : !S (f (x1) f (x2 )) = !(f (f (x1) f (x2 ))g ) 6 !(f (x1 x2)g) = = !S (x1 x2). 4 & , & &' ' ' !( ( & &'( . , F $ 2 R, . . (f f )(R) R !& $ 2 B . 4 $ - Y & F^ . , ' 2 ' ! . 2. ! X F , ) Y F^ .

30

. .

. y1 y2 | ' ' Y , Z | , ! ( X . A f^ | & F^ , f | ! $ & F , &' f , &!( Z , ! Z , !! f^ (y1 ) f^ (y2 ). S |

!( ( &

&'( F X . 4 $ &' &' 2 S , Z & 2 ' Z . B, ^ ^ (f^ (y1 ) f^ (y2 )) 6 ^ (y1 y2 ), . . & F^ . 76 , 2 ! Y . 3. 5 ( & $ $ . 6 ' (

& , ( $

& '( $ . 3. . . S , A. % S !! ! S = f0 1 : : :g, & . ! + 1 T = fT0 T1 : : :g S , ! ' 6 T ! : T1) 2 T > 9 T2) T | 9 T3) T \f : < g $ & '( T . 7 , T ' . , $ , T0 = S , T ' ( < 6 . A 2, , = + 1 $ & T n f g, T = T . A $ & , T = T n f g. %' T1), T2), T3) . T | ' . 7& $ P = T . , '/ .' '3

! " 39] !. '3 ! 3 # /. ? / /. " ?. 4. @! 1 ! ., ' 1 1" ! - " - .

6.

4 0, ! ) g 6 a (C). '(f ; fm ). ': f 0 +': f 00 & f 0 f 00 2 Fbc (T)% 6) ('e(fn ) j n 2 N) " 'ef & (fn 2 Fbc (T ) j n 2 N) & f 2 Fbc (T), (fn j n 2 N) " f F(T ). . % # 1) ) . 2) ': f supf'h j h 2 Z ^ h 6 f g 'ef inf f'g j g 2 Y ^ ^ g > f g. K! !! h 6 f 6 g, # 6) ''/ 5 'h 6 'g. ; ': f 6 'g ': f 6 'ef. 3) K! !! 'e(;f) inf f'g j g 2 Y ^ g > ;f g, f > ;g 2 Z )& ': f > '(;g). # 5) ''/ 5 '(;g) = ;'g. ': f. % # 4) 1 # 3) ''/ 5. 5) 'e(f 0 + f 00 ) inf f'g j g 2 Y ^ g > (f 0 + f 00 )g, 0 'ef inf f'g0 j g0 2 Y ^ g0 > f 0 g 'ef 00 inf f'g00 j g00 2 Y ^ g00 > f 00 g. K! !! g0 +g00 > f 0 +f 00 g0 +g00 2 Y, # 3) ''/ 5 'e(f 0 +f 00 ) 6 '(g0 +g00) = = 'g0 + 'g00. ; 'e(f 0 + f 00 ) ; 'g0 6 'g00 )& 'e(f 0 + f 00 ) ; 'g0 6 'ef 00 , 'e(f 0 + f 00 ) ; 'ef 00 6 'g0 )& 'e(f 0 + f 00 ) ; 'ef 00 6 'ef 0 . 6) K! !! ('efn j n 2 N) " 'efn 6 'ef b - n, a sup('efn j n 2 N) 6 b. P!' " > 0. K - !. n gn0 2 Y, ! ) gn0 > fn 'efn + "=2n > 'gn0 > 'efn . K! !! f 2 Fbc (T), f 6 u - ! u 2 Cc (T). '' gn00 gn0 ^ u 2 Y. # 3)

40

. . , . .

''/ 2 g 2 Y, ! ) g = sup(gn00 j n 2 N). K! !! gn00 6 gn0 ,

# 1) ''/ 5 'gn00 6 'gn0 < 'efn +"=2n . 2' , gn00 > fn ^ f = fn . '' gn sup(gi00 j i = 1 : : : n) 2 Y. K (gn j n 2 N) " g. # 4) ''/ 5 ('gn j n 2 N) " 'g. ' !3, ) 'gn 6 'efn + " ; "=2n . =- n = 1 'g1 = 'g100 6 'ef1 + "=2 = 'ef1 + " ; "=21. 1 ' # / ) gn + gn00+1 = gn ^ gn00+1 + gn _ gn00+1 . K! !! gn ^ gn00+1 > > gn00 ^ gn00+1 > fn ^ fn+1 = fn , '(gn ^ gn00+1 ) > 'efn . 2' , gn _ gn00+1 = = gn+1. 4 1" 'gn + 'gn00+1 = '(gn ^ gn00+1) + 'gn+1 > 'efn + 'gn+1 . % ", 'gn+1 6 ('gn ; 'efn )+'gn00+1 6 " ; "=2n +'gn00+1 < " ; 2"=2n+1 + + 'efn+1 + "=2n+1 = 'efn+1 + " ; "=2n+1. - ! !1' , )' 'g = lim('gn j n 2 N) 6 lim('efn +" ; "=2n j n 2 N) = lim('efn j n 2 N)+" ; lim("=2n j n 2 N) = = a + ". 2' , f = p-lim(fn j n 2 N) 6 p-lim(gn00 j n 2 N) 6 6 p-lim(gn j n 2 N) = g 2 Y )& b 'ef 6 'g. 4 1" b 6 a + ". K! !! " 1 ", b 6 a 6 b. M '' Fbc (T ) '. Xbc' ff 2 Fbc (T) j 'ef = ': f g. ; ' /# !3 '^ Xbc' , . 'f ^ 'ef = ': f. = ' ' " X' . % '" / .# !1/ - - '. 1. T | ' 2 2 (Cc (T ) )+ .

1) X' = ff 2 Fbc (T) j 8" > 0 9g 2 Y 9h 2 Z (h 6 f 6 g ^ 'g ; 'h < ")g% 2) X' | % 3) Y Z X' % 4) '^ ! ! '% 5) '^ - X' . . 1) ;1)' )" 1) )1 X. 'ef inf f'g j g 2 Y ^ g > f g ': f supf'h j h 2 Z ^ h 6 f g. =- f 2 Fbc (T ) " > 0 g 2 Y h 2 Z, ! ) h 6 f 6 g 'ef + "=2 > 'g ': f ; "=2 < 'h. D f 2 X' , 'g ; 'h < ('ef + "=2) ; ; (': f ; "=2) = " )& f 2 X. ;, f 2 X " > 0, 'ef 6 'g ': f > 'h ! 'ef 6 'g < 'h + " 6 ': f + ". J), 'ef 6 ': f. L "1 # 2) 1 .- 1, 1! )', ) 'ef = ': f, . . f 2 X' . 2) " f 2 X' a 2 R. K # 3) 1 .- 1 'e(;f) = ;': f = ;'ef = ': (;f). J), ;f 2 X' . D a > 0, 'e(af) = a'ef = a': f = ': (af). D a < 0, 'e(af) = = (;a)'e(;f) = (;a)': (;f) = ': ((;a)(;f)) = ': (af). J), af 2 X. " f 0 f 00 2 X' . K # ' 2) 5) 1 .- 1 ': (f 0 +f 00 ) 6 6 'e(f 0 +f 00 ) 6 'ef 0 + 'ef 00 = ': f 0 +': f 00 6 ': (f 0 +f 00). % ", f 0 +f 00 2 X' .

41

3) " g 2 Y. K 1 'g supf'f j f 2 Cc (T) ^ f 6 gg , ) - " > 0 f 2 Cc (T ) Z, ! ) f 6 g 6 g 'g ; " < 'f = 'f. J), g 2 X' . ? ), h 2 Z, 1 'h = inf f'f j f 2 Cc(T) ^ f 6 hg , ) - " f 2 Cc (T) Y, ! ) h 6 h 6 f 'h + " > 'f = 'f. J), h 2 X' . 4) " f 2 Cc (T) Y X' . K 'f ^ = 'ef = 'f = 'f. D f 2 X' f > 0, '^f 'ef > 'e0 = 0. " a 2 R f 2 X' . D a > 0, '(af) ^ = ': (af) = a': f = a'f. ^ D a < 0, '^(af) = 'e((;a)(;f)) = (;a)'e(;f) = a': f = a'f. ^ " f 0 f 00 2 X' . K 1 !1" # 2) , ) '(f ^ 0 + f 00 ) = ': (f 0 + f 00 ) = ': f 0 + ': f 00 = 'f ^ 0 + 'f ^ 00 . 5) K! !! '^ = 'ejX' , # 6) 1 .- 1 (fn 2 X' j n 2 N) " " f 2 X' )& ('f ^ n j n 2 N) " '^f. K! !! X' - -- !/' 0&)/' ', f(t).

42

. . , . .

% # /, g0(t) = (u(t) ; f(t)) + f(t) ; "=4 > f(t) ; "=2 0 h (t) = (u(t) ; f(t)) + f(t) + "=4 < f(t) + "=2. K! !! !3- f -

) !' !/' ', !' ! '. C ) a b, ! ) a (C) 6 f 6 b (C). ;1)' (C) )1 x '' )/ !3 !' !/' -' v (ax _ g0 ) ^ bx w (ax _ h0 ) ^ bx. K f(t) ; "=2 < v(t) 6 f(t) 6 w(t) < f(t)+"=2 - t 2 T 0 6 w ; v 6 "x. P K! !! v 2 St(T A(T G )), v = (ak (Ak ) j k 2 K) - !/ !)/ !!3# (ak 2 R j k 2 K) (Ak 2 A(T G ) j k 2 K). 4 , ) A(T G ) | , '. )", ) S '. Ak ! - ak )/ -. K (Ak j k 2 K) = coz v C. % ", !. '. Ak - -- !' !/'. '' 7 (Ak ) 2 X' . J), v 2 X' . ? )/' 1' w 2 X' . 41"'&' 1 " > 0. !1' /0 - " =(3'(x)) ^

&' v w 1 X' . 4 .- 1) 1 '/ 1 g0 g00 2 Y h0 h00 2 Z, ! ) h0 6 v 6 g0 , h00 6 w 6 g00 , 'g0 ; 'h0 < =3 'g00 ; 'h00 < =3. @, ) h0 6 f 6 g00 . 2' , 'g00 ; 'h0 = ('g00 ; 'h00 ) + ('h00 ; 'g0 ) + ('g0 ; 'h0 ) < 2=3 + 'h ^ 00 ; 'g ^06 6 2=3 + 'w ^ ; 'v ^ = 2=3 + '(w ^ ; v) 6 2=3 + "'x ^ = . % ", f 2 X' . M '' " #/# ."/# !3 '^ '. Sc (T) !.', ) 1=ng D0 !3 hn (Kn ). @, ) !' !/. K! !! (Kn j n 2 N) " D0 , (hn j n 2 N) " g0 . 4 ))# - / !3 '^ '/ )' ('h ^ n j n 2 N) " 'g ^ 0 . 0. '' 8 - An !' ! '. Ln, ! ) Ln TAn '( (A ^ n n Ln )) < "=2n+1 . ''S !' !/ '. KnS (Li j i = 1 : : : n) An . K! !! An n KnP= (An n Li j i = 1 : : : n)

P(Ai n Li j i = 1 : : : n), '^( (AnPn Kn )) 6 '( ^ ( (Ai n Li ) j i = 1 : : : n)) = = ('( (A ^ i n Li )) j i = 1 : : : n) < " (1=2i+1 j iT= 1 : : : n) 6 "=2. '' !' ! '. K (Kn j n 2 N) !3 h (K) hn (Kn ). K! !! (Kn j n 2 N) # K, (hn j n 2 N) # h. 4

))# - / !3 '^ )' ('h ^ n j n 2 N) # 'h. ^ % ", n, ! ) '^h + "=2 > '^hn. 4 1" ' (A ^ n n K) = ' (A ^ n ) ; 'h ^ < ' (A ^ n ) ; '^hn + "=2 = ' (A ^ n n Kn ) + "=2 6 ". J), !3 '^ - -- ."/' !' !/'. =, - # )! t 2 T !' !- !/!" D. '' !3 g (D) 2 X' . D f 2 X' ^ 6 '^g )& j'f ^ j 6 'g ^ < 1. J), '^ jf j 6 g, ;'g ^ = '( ^ ;g) 6 'f - -- !" )/'. (!3, # ))# - / 1 '/ 1. K!' 1', !3 '^ - -- !'. " " - -- #/' ."/' !' !3' Sc(T ), . ' !3 '. =!.', ) = '. ^ %) !.', ) (D) = ' (D) ^ - !' ! !/ '. D. K! !! - -- ."/' !' !/', - " > 0 !' ! '. C D, ! ) (D n C) < ". ;1)' (D) (C) )1 g h . % ", g < h + ". '' 1'! '. F T n D. K! !! T | ! '. C | !' !, / - !3- f T, !) 0 6 f 6 1, f(t) = 0 - t 2 F f(t) = 1 - t 2 C. @, ) g 6 f 6 h. 0 " =(x), x | !3- 1 !1" .- 2. =P " 1"'&' !3 v w 1 !1" .- 2. K v = (ak (Ak ) j k 2 K) G ) j k 2 K). % ", - !# !)# !!3 P P ^(Ak k2) Aj kc(T v = (ak (Ak ) j k 2 K) = (ak ' (A 2 K) = 'v. ^ ? )/' 1', w = 'w. ^ K! !! v 6 f 6 w 0 6 w ; v 6 "x, f 6 w 6 v + "x = = 'v ^ + 6 '^f + 'f ^ 6 '^w 6 'v ^ + "'x ^ = v + 6 f + . % ", jf ; 'f ^ j 6 . K! !! 1 ", f = 'f. ^ M % 0. ;1)' a' )1 . 41"'&' / g 2 Y " > 0. '' '. Lg ff 2 Cc (T ) j f 6 gg. K 1 'g = sup('f j f 2 Lg ) g = sup(f j f 2 Lg ) , ) u v 2 Lg , ! ) 'g ; "=(2a) < 'u g ; "=2 < v. '' !3 w u _ v 2 Lg . K j(a')g ; gj 6 ja'g ; a'wj + jw ; gj < aj'g ; 'wj + + "=2 < ". L1 1 " " , ) (a')g = g. % 0 )/' 1' !1/ -, ) (a')h = h - h 2 Z. " " f 2 Sc (T ) " > 0. '' '. Zf fh 2 Z j ^ = sup(h j Zf ) ! ., !! h 6 f g. K 1 'f ^ = ': f = sup('h j h 2 Zf ) f ^ /0, , ) (a')f ^ = f. ;1)' '0 + '00 )1 . " g 2 Y " > 0. K 1 ' 0 g = = sup('0f j f 2 Lg ), ' 00g = sup('00 f j f 2 Lg ) g = sup(f j f 2 Lg ) , ) u v w 2 Lg , ! ) ' 0g ; "=3 < '0 u, ' 00 g ; "=3 < '00v g ; "=3 < w. '' !3 x u _ v _ w 2 Lg . K j(' 0 + ' 00)g ; gj 6 j' 0g ; '0xj + j' 00g ; '00 xj + jx ; gj < ". % ", (' 0 + ' 00 )g = g. ? )/' 1' !1/ -, ) ('0 +'00)h = h - h 2 Z. " " f 2 Sc (T ) " > 0. K '^0f = ': 0 f = sup('0h j h 2 Zf ), ^ = sup(h j h 2 Zf ). K! ., !! /0, '^00f = sup('00h j h 2 Zf ) f 0 00 ^ , ) ('^ + '^ )f = f. ' ", ) . P0 1 - / -! / 3/. " ' 6 . 41"'&' g 2 Y '' '. Lg ff 2 Cc(T) j f 6 gg. K 1 'g = sup('f j f 2 Lg ) g = sup(f j f 2 Lg ) , ) 'g 6 g. 41"'&' " f 2 Sc (T ) '' '. Yf fg 2 Y j ^ = inf(g j g 2 Yf ) , g > f g. K 1 'f ^ = 'ef = inf('g j g 2 Yf ) f ^ K!' 1', P0 - -- '/'. L1 ! P0 ) 'f ^ 6 f. " , ) P0 - -- 1/'. % ", P0 - / -! / 3/. 4 .- 3.6.1 1 16] . P0 ' 0 5! # P 1 A Cc (T) B Sc (T )4 , ! ) P' = P0('+ ) ; P0(;'; ). K! !! P0 - / -! / 3/, '1 ^ '2 = 0 )& P'1 ^ P'2 = P0 '1 ^ P0'2 = P0('1 ^ '2 ) = 0. # 14E(b) 1 12] P - -- 0&)/' #/' '. D 2 B, = + + ; . % 1 1 '/ 2 + = = P0'0 ;; = P0'00 - !/ '0 '00 2 A+ . '' !3 ' '0 ; '00 2 A. K = P0'0 ; P0 '00 = P '0 ; P'00 = P '. 9 1), ) P - -- 5! /'. K!' 1', P " ! /# 0&)/# #/# . 4 (L) > 4A , L RE (L) 6 "(L). 0 * ) "(L) (L) ) A , * * ) ) L = K, K4, T, S4, KB, GL, Grz 31] "(L) = 1 , * (L) = 1, , , "(S5) = (S5) = 16. 7 * (L) ) ( . 35]).

55

4.1. "(L) (L)

0 * * "(L) (L) ) , ) 3. J # 3.4 ) . . 9 M

" L ( * : M ,! L), # r, * M = L(r). U* , * * (L) * M, * ) L. 4.1.1. () N (N = I, II, III) $

! ! ! ) . . : N = I 0 * . N = II. L II r ( , ) * 0 ), 0 3.5 L0 II # (p q), * L( (( ) r)) = L0 . N = III. : , , * L?()> ,! L():> . J

L():> ).. x _ y, 0 ` ?_ >. 1 r, ).. :x _ y. J 0 , * , r ,

(p) (:p ^ : ?) _ (p ^ >), L():> ` r? $ : ?, ` r> $ >, *, ` :r? _ r>. &0 L():> (r) = L?()>. Y* III ).. , * kk = 3. 4.1.2. () N (N = I, II, III) $

! N 0 > N . . C* N = I * .

4.1.1 / ,! * N = II, III N, * - N 0 = N + 1. N = II. r = > L?(): , *, L?(): (r) = L?> 3.5. N = III. & L | III. & 3.4 : L = E f p $ rpg, *# r | . 9 , * ).. r r L?():> ).. L. & 3.2 * L?():> (r) = L. . Q1 Q2 L, ) , L, A , L ) * r , (p) (p ^ Q1) _ (:p ^ Q2), >.

56

. .

4.1.3. * N (N = II, III, IV) $ ! N 0 < N . . / ,! *

N 0 = N ; 1 * / ) . & N = II * . N = III. L?> 0 ( ) >, *, 3.5 L?> * / I II. N = IV. L?(): ) , *, , ).. ) >, 0 L?():> 6,! L?(): . 4.1.4. + L | N = I, II, III, IV, (L) = 4 10 14 15 "(L) = 4 16 64 256

. (L)

). L 0 b(( ) ? >)A 256 ) . # b1 b2 b (b1 $ b2). U* , * L ` b(p ? >) , b( x y) > (x y) 2 f? >g ) ) ( . 3). D*, L ` b1 $ b2 , b1(p x y) b2 (p x y) * / ) ) f( 0 1) 2 f? >g3 j (0 1) = ?g

2 (4 ; k k), *# k k = N ( . * 3.4.2). U "(L) = 256=22(4;N ) = 22N . 4.2.

0 * ) "(L) (L) ) * ) ) . . # ( . 31, c. 4])

, (A1) * L (A2) (A ! B) ! ( A ! B) ( ) MP, Sub Nec ( ) A ` A: ( * * K. J / , *

. 4.2.1. + ) L ) K

MP Sub,

L | ( L RE) , , L | ( L Nec).

57

0 * , K ) : (A3) p ! p ( ) (A4) p ! p () (A5) p ! 3p (

*) (A6) 3p ! 3p ( ) (A7) ( p ! p) ! p () 9# ) (A8) ( (p ! p) ! p) ! p () 8 *): J 31, . 5]: T = K + (A3)A K4 = K + (A4)A S4 = T + (A4)A KB = K + (A5)A S5 = S4 + (A5) = T + (A6)A GL = K + (A7) | 8 {9# A Grz = K + (A8) | 8 *. & 0) 31, . 5, 12], , * L 0 , S5, "(L) = 1, "(S5) = 16. 0 0 * (L). 4.2.2. + L | K L GL, "(L) = (L) = 1. . & "(L) > (L), * , *

(L) = 1. U : r1 = ( )A rn+1 = ( ) ^ 3rn n > 1: (1) & , * rn L *. & N > m > 1 | *, p1 : : : pN | * , N = f1 : : : N g. = : ANm

_ j 2N

pj !

_

J N jJ j=m

_ j 2J

pj :

4.2.3. + m > n, ANm 2 K(rn) N > m. . 31, . 5] K:

K ` A , A * ) *) /) . & (W R j=) |;

, x1 2 W. & W pj A 0 * , * x1 j= rn : ( j 2N * ) 0 x2 : : : xn 2 W, * x1 R x2 R : : : R xn, 8i = 1 : : : n 9j = j(i) 2 N xi j= pj . 1, ; W pj . J N , jJ j = m, * J fj(1) : : : j(n)g, * x1 j= rn j 2J

4.2.4. + m < n, ANm 2= GL(rn) N > n. . 9 GL *) -

) ) / 31, . 5]. = W = f1 : : : ng * < ) *)A

58

. .

i j= pj , i = j, 1 6 i 6 n, j 2 N . 1 (W n N > maxfm ng, *, L(rn) 6= L(rr ) n 6= r. * K K4 GL 31, . 1] , * (K4) = 1. = - . (. 31, . 10]), :, ::. 4.2.5. * ! GL $ , -, )# : . . 9 r m 3Q m : 3Q, m > 1, Q | . 1 GL ` m 3A $ m ?, r 0 GL m ? : m ?, *, GL(r) = L?> 3.5. 310], GL

: GL( n ) = GL n > 1. C , GL 7 *) - .: ( ) : : : 3 : . = , (A3). 2 ) 4.2.2A , ) (1) 0 ( ). U . 4.2.6. + L | K L Grz, "(L) = (L) = 1. . :* , * (L) = 1. # : r01 = ( )A r0n+1 = ( ) ^ 3(:( ) ^ 3r0n) n > 1: (2) & , * L(r0n ) *. 7 ANm , 4.2.2. 4.2.7. + m > n, ANm 2 K(r0n) N > m.

4.2.3 *# , ; W p * x1 j= r0n j * , * ( j 2N * ) 0 y2 x2 : : : yn xn 2 W , * x1 R y2 R x2 R : : : R yn R xn 8i = 1 : : : n 8j 2 N yi = 6j pj 8i = 1 : : : n 9j = j(i) 2 N xi j= pj : r

4.2.8. + m < n, ANm 2= Grz(r0n) N > n.

59

. , * Grz *) ) )

*) / 31, . 12]. J W = f1 : : : 2n ; 1g * 6 / j= : i j= pj , i = 2j ; 1, i 2 W, j 2 N . 1 (W 6 j=) | Grz 1 =6j tr n (ANm ). ? )

. 1 * K T S4 Grz ( . 31]), 4.2.6 (T) = (S4) = 1. KB rn (1) n > 1 0 r2 A * (2). 1 . 4.2.9. + L | K L KB, "(L) = (L) = 1. . 1 "(KB) = 1, "(L) = 1. = r00n, ) 00 00 00 1 (p) = p ^ 3pA n+1 (p) = p ^ 3(p ^ 3(:p ^ 3(:p ^ 3 n(p)))) n > 1 (3) ANm , 4.2.2. 9 , * m > 2n, ANm 2 K(r00n) N > m. & KB

*) / , , * m < 2n, ANm 2= KB(r00n) N > 2n. C , L(r00n ) * (L) = 1. J , S5. 2 / (W R) / R = W W. 4.2.10. "(S5) = (S5) = 16. . 9 S5 0 ) ( 0 )) : (i) r, :r, r:, :r:, r | 0 = ( ) ! A (ii) r, :r, r | 0 ( ) = ^ 3A (iii) r, :r, r | 0 ? = _ :. : * , * 0 ! . D /, * 0 0 ( ) , 0 0 ! * . & , * (S5) = 16. ( (iii) $ , ) ( S5) :p $ p. ( (ii) $ : ) :p $ : p. 9 ? > ( ) : * ) . r0

60

. .

: , ( p $ p) 2 S5(:) n S5()A p 2 S5() n S5(:). D*,

(ii) (iii) *. J * (i). 4.3.

!

D 35] . & T U | * , T * . = * , ) * , T . 1 ) , ) U ,

. ( 0) RE , *, / . &0 4.3 RE. J ) ) : D = GLf: ? ( p _ q) ! ( p _ q)g | : A S = GLf p ! pg | C . # * Fn n+1 ? ! n ?, n 2 ! = f0 1 : : :g. 36] , * * : GL = GLfFn j n 2 g D = D \ GL GL = GL

_

;

n=2

;

:Fn S = S \ GL ;

!, ! n *. _ ! n *, GL D S GL A , D! = D, S! = S, GL! = Fm. = * C S, ) * ) . U/ S, 37]. . ! M = (W j=) ( W | , |

** W ), 0 r %""

b, * (1) fx 2 W j r xg | * ( | / , / )A (2) fx 2 W j x rg * (! + 1) A (3) 0 W rA (4) b | / 0 WA ;

;

61

(5) ) *) fx 2 W j x rg. N A

M,

* : M b j= A. 4.3.1 (%7]). S ` A , A ! ! . & S, , * (2) 0 SA , "(S) = 1. 4.3.2. (S) = 1. . = (2) ANm , 4.2.2.

4.2.7, m > n, ANm 2 K(r0n ) S(r0n) N > m. & m < n. & ) M = (W j=),

tr n (ANm ) N > n. & W = fbg V , V = f: : : x;2 x;1 x0 x1 y2 x2 : : : yn xng, W * / (! +1) , 0 b | *, 0 ) V : : : : x;2 x;1 x0 x1 y2 x2 : : : yn xn . U / : b j= p1A x;i j= p1 ) i > 0A xi j= pi i = 1 : : : n. 1 M | ) ) 0 r = x1 , b = 6j tr n (ANm ). 4.3.3. + L | ( RE), $ K L S, "(L) = (L) = 1. 1 , L " 35], L S, "(L) = (L) = 1. ? 36] , * , S ( " 35]), * GL . & , * L "(L) < 1 (L) < 1. & k = 3k 1) = fn 2 ! j n > kg, k > 0. = * GLk = GLk . U* , * GLk = GLf k ?g. . & ) F # * , ) ) 0 ) F . H F (k) * , * F ) )

k ?. # * (r )(k) = r (k) . ' deg(r ) r # (p), deg(?) = deg( ( )) = 0A deg(r1 ! r2) = maxfdeg(r1 ) deg(r2)gA deg( r) = 1 + deg(r): r

r0

;

;

;

;

62

. .

4.3.4. GLk ` F $ F (k) ! F . . :* k , * GL ` k (k ) ;

`

? ! (F $ F ).

4.3.5. "(GLk ) < 1, (GLk ) < 1. . J GLk r 0 r(k), ;

;

;

/ * * 0 ) * , *, "(GLk ) < 1. : , 35] , * GLk Nec, *

4.2.1 RE, , *

(GLk ) 6 "(GLk ) < 1. # ! * . C k > 0, * (! n ) fn 2 ! j n < kg, 0 3k 1) GLk GL . C , "(GL ) 6 "(GLk ) < 1. U, , *, * (GL ) < < 1, GL , , RE. 1 , , * L = GL r * r(k). & A 2 L(r), L ` tr (A). 1 L GLk ,

4.3.4 L ` (tr (A))(k) . 9 ( A r), * (tr (A))(k) (tr (k) (A))(k) , L ` (tr (k) (A))(k) ,

4.3.4 L ` tr (k) (A), * A 2 L(r(k)). 2 , 0 L(r) = L(r(k)). 1 , (GL ) < 1. ;

;

;

;

;

;

;

;

;

;

;

;

r

r

r

r

r

r

;

5. #

0 , MP Sub. 4.1

(,!)

) A * 0 / ) . D 0 / ) ) ), * ) ) ) ( . ) 4.2 4.3). 7 ) ) * ) , * ) ) ). # * . . 9 M

" L, r, * M L(r). U/ , , 38] ) ) 9 ( * , * M L, 38] - L M- .).

63

5.1. () !

.

. & L | . ? , * ) 39], , * L ` : ?, L L() , * L L> . J L() L> * . C

5.2 5.4 )

, , * , ) * . 5.2. M L(r) L RE, 3M] L(r), 3M] | , )# M RE. 5.3. + 0 2 !, GL GL. . &

4.2.1 3GL] Nec, 0 : ? ( ? ! ?), GL 0 ?. D*, 3GL ] * GL. 5.4. + M L L ) !- L? L() L: L> , M ) ! . . _ L Lrj , j 2 f0 1 2 3g ( * . 3), L(r) Ltr j (r) . : * # 35] t(L) L. & (W ) | * r. (%5]). _ x | (W ), d(x) = 0A * d(x) = maxfd(y) j x yg. ( M = (W j=) # . ' A t(A) = fn 2 ! j M n r, * M r =6j Ag. ' t(L) S L O ) , ) L: t(L) = t(A). A2L = , ) GL. : L 0 : L L> , L 0 , 0 2= t(L)A , L L? L() L: . U . 5.5. + L M ) GL, 0 2= t(L) 0 2 t(M),

M L. . & L L> , M

L? L() L: L> . : * . r

64

. .

5.6. % # , $ M ,! L: (i) "(M) 6 "(L) (M) 6 (L). (ii) $ M $/ $ L. (iii) const(M) 6 const(L), const(L) $ 0 L. (iv) L RE, 0 ) M . 5.7.

(i) * $

! . (ii) + L RE ) > : ?, GL 6,! L. (iii) + L ) GL ) ! $

, GL 6,! L. (iv) GL $ , ! GL, ) , ! 3k 1), k > 0.

.

(i) C

5.6,(i) 4.3. (ii) L ) . (iii) ? 31, . 7], * GL 0 n ?, n 2 GL ` ` n ? $ n+1 ?, 0 const(GL ) < 1. D*, const(L) < 1, const(GL) = 1, 0 GL 6,! L

5.6,(iii). (iv) 35] , * Nec ( RE, *

4.2.1) GL GL , ) 3k 1) k > 0. . N A p, ) 0 A ) . N A # , A , A B. & p = (p1 : : : pn) | , A , ( * ) 0 : _ (p ^ B ) A$ (]) ;

2f?>gn

B | . . 7 , * L , A L ` A L ` B 2 f? >gn,

65

B | (]). : , L , ) ) . 5.8. L | , M | , )# $ (A5) p ! 3p, M L(r) ! r. L(r) ! L() , L> , L()>. . :* , * L()> L(r). = rp. _# (]) : rp $ ((p ^ Qp) _ (:p ^ Q0 p)): 9 M , *, L ` r(p ! q) ! (rp ! rq). & (]) 0 L, * : (a) L ` Q(p ! q) ! (Q0 p ! Q0q)A (b) L ` Q(p ! q) ! (Q0 p ! Qq)A (c) L ` Q0 (p ! q) ! (Qp ! Q0q)A (d) L ` Q(p ! q) ! (Qp ! Qq): Y* (p ! 3p) L ` (:Q0:p ^ Q:r:p) _ (Q0:p ^ Q0:r:p), 0 O ) ) ( :p p): (e) L ` Q0 p ! Q0:rpA (f) L ` Q0 p _ Q:rp: , M Nec, 0 r- M L (g) L ` A, L ` QA. 1 L ` Q(p ! p), , p q (b), * (h) L ` Q0 p ! Qp: (e) (f) (i) L ` Q0 :rp _ Q:rp (h) (j) L ` Q:rpA r 0 * (k) L ` Q((Qp ! :p) ^ (Q0p ! p)):

66

. .

? (d) (g) , * L(Q) , 0

: (l) L ` Q(A ^ B) ! QA , (k) * (m) L ` Q(Qp ! :p) (d) (n) L ` QQp ! Q:p: ? (m) (g) (o) L ` QQ(Qp ! :p)A 0 (n), * (p) L ` Q(p ^ Qp) (q) L ` Qp: 1 , (r) L ` rp $ (p _ Q0p) * (s) L ` r? $ Q0?: 1 (a) * L ` Q0p ! Q0 q, *, (t) L ` Q0 p $ Q0?: ` (s) (t) (r) (u) L ` rp $ (p _ r?) L()> ( . 3.4) * : L()> L(r). H * ) , . 5.9. ' GL . . = (]) - A. : GL 6 ` B 2 f? >gn,

67

GL 31, . 84] | * r, * r =6j B . ` r =6j B ) * r, 0 / j= 0 * , r j= pi , i = >. 1 r = 6j A, * GL 6 ` A.

5.10. + K 6 ` A, !/ ( (W R j=) 0 r 2 W , $ r =6j A # x 2 W , $ x R r. . & K 31, . 5] M = (W R j=), * r = 6j A r 2 W. & # M0 = (W 0 R0 j=0). & W 0 = W fr0g, r0 2= W. U/ R0 : 0 ) W n frg / R0 R, x y 2 W n frg, x R0 y , x R yA M0 r r0 * W n frg, * r ) , r R0 x , r0 R0 x , r R x x 2 W n frgA M0 0 r0 ) 0 W n frg, ) r ) , x R0 r0 , x R r x 2 W n frgA r R r, r R0 r0 R0 r0. U* , * x 2 W 0 x R0 r. & / j=0 / j= W 0 : r0 j=0 p , r j= p p. 1 , * F : r0 j=0 F , r j= F 8x 2 W (x j=0 F , x j= F): C , r =6j 0 A. 5.11. ' K K4 . ( *#

5.10, K4)

5.9. 5.12. + L | X | ) , LX | . . V : A LX ` A 0 , * L ` ; ! A * ; X. ? (]) A, *

^ _ p ^ ; ! B : ;!A $ 2f?>gn V V ? LX ` A L ` ; ! A, L ` ; ! B ^

2 f? >gn, *, LX ` B .

68

. .

5.13. 1 GL , D , GL , !, ! n $ . J * ; = ( ) ^ . 5.14. L ) , L(;) L L(;) L. L(;) = L + f p ! pg. . * () * . : * : L(;) L1 , L1 = L + f p ! pg. & L1 ` p $ ;p, L1 ` A $ tr; (A) A. & A 2 L(;), tr; (A) 2 L, tr; (A) 2 L1 , *, A 2 L1 . 31, . 12], GL(;) = Grz. ?

, * K(;) = T, K4(;) = S4. J B: B = T + (A5), (A5) |

*: p ! 3p. 5.15. ' KB, B, S5, ) 2 S5, )# L()> , $ K, T, K4, S4, Grz, GL , D , GL , !, ! n $ . . ? 5.8 , * ) , )

*, / , * ) ): 0 L() , L> L()> . : , 311] , * / S5 . &0 K, K4, GL , D GL ) . : T, S4 Grz / ,! , * T ,! K, S4 ,! K4 Grz ,! GL. & # , ) GL * . 5.16. GL ,! GL $ ) !. .

310] GL: GL( n ) = GL n > 1. # k > 1, *

30 k) = fn 2 ! j n < kg. = *. 0 2= . & , * GL( k ) = GL( k ). * () * . & A 2 GL ( k ), GL ` tr k (A). 1 t(tr k (A)) 30 k). U* , * GL- *) ,

/ * k, tr k (A) tr> (A) . J * * GL L> tr> (A) 0 ( , tr k (A)) . U GL ` tr k (A), *, A 2 GL( k ). 0 2 . 0 * , * GL ( k+1 ) = GL( k+1 ). D * V* (). U * F = Fn. U* , * GL ` B , GL ` F ! B. ;

;

;

n2

69

& A 2= GL( k+1 ), (W j=) r, * r = 6j tr k+1 (A). _ d(r) < k, / (W ) ) 0 xk xk;1 : : : xd(r) = r ( * d(xk ) = k) / j= , * *) xk r . 1 *) xk r tr k+1 (F ), * xk =6j tr k+1 (A). 1 , *, * d(r) > k, r j= F , t(F ) = 30 k). C , r = 6j F ! tr k+1 (A), *, GL 6 ` F ! tr k+1 (A) A 2= GL ( k+1 ). Y _. y. J 1. 9. {

) * .

1] Boolos G. Logic of Provability. | Cambridge: Cambridge University Press, 1993. 2] . . | ., 1974. 3] Makinson D. There are innitely many Diodorean modal functions // Journal of Symbolic Logic. | 1966. | Vol. 31, no. 4. | P. 406{408. 4] Sugihara T. The number of modalities in T supplemented by the axiom CL2 pL3 p // Journal of Symbolic Logic. | 1962. | Vol. 27, no. 4. | P. 407{408. 5] $%& '( ). *. + ',- -, '&.%/01- ./ '& // 2.(. $* ))). ) % '& '. | 1985. | 3. 49, 4 6. | ). 1123{1155. 6] 5 '6 ( 7. 8. + 9: ;%;.:,- ./ '& // 2.(. $* ))). ) % '& '. | 1989. | 3. 53, 4 5. | ). 915{943. 7] Visser A. The provability logics of recursively enumerable theories // Journal of Philosophical Logic. | 1984. | Vol. 13. | P. 97{113. 8] Zeman J. Modal systems in which necessity is 6. . $. )( &( & %&(& ( - ./ '& // 3 ., 8 ? 0.. 9. ; '&. . (, &. 1986. | ). 4. 11] Scroggs S. J. Extensions of the Lewis system S 5 // Journal of Symbolic Logic. | 1951. | Vol. 16, no. 2. | P. 407{408. ' ( ) 1997 .

. .

. . .

517.588+519.68

: , , !" " "# , $% "&.

'( ) *

* ) ) (" "!$ ( ( ( &" &+ ( !" " "# * (, %+* !() ! " "$

. , !" "$ &- % ( !& ) &"$. "

* ( "! * !"$(,

% (%+) & . ) " "%&* * (, " ( !( ! !. /, ! ("$ $ "

$ ( * . ), "!

* (

) !+* &* (, &(% (# !% " "# ) | ( " . ! !! ) ) * | ( # $ ( (

$% " "

* ( * &( ) & -" ) " ! .

Abstract A. W. Niukkanen, Analytical continuation formulas for multiple hypergeometric series, Fundamentalnaya i prikladnaya matematika, vol. 7 (2001), no. 1, pp. 71{86.

Applying canonical forms of multiple hypergeometric series along with the use of the operator factorization method makes it possible to obtain, in explicit and most general form, the analytical continuation formulas directly applicable to arbitrary series having the Gaussian type (2==1) with respect to one or several arguments. The formulas help us to unify a great number of particular formulas scattered throughout the literature. Moreover they give us a complete set of relations for any non-standard series provided that it pertains to the Gaussian type with respect to at least one of its variables. Due to simplicity and universality of the basic relations /"# "( ), (

* &) () * (, ("% !"$(

) &, # , ("

) "$ ). !"!&"6

( "# ), "!" #! 7) ! "$ * "( ) ( 97{01{00317 01{01{00380). , 2001, 7, 9 1, . 71{86. c 2001 ! " # $%&, ' () *

72

. . there arises an importantpossibilityto implementcomputer-aided analysis of numerous repeated transformations with respect to di:erent arguments of the series and to join these transformations with other important types of transformations. This possibility may have signi;cance for mathematical analysis, mathematical physics, computer algebra and theoretical chemistry.

1.

1] 2{5] ! !"$ , & !'$$$ (2==1) ' . N F(x1 : : : xN ) ' xn $ ' $ "$ (p==q) ! Fqp, '"! N F ' $ N F $ &" xn . + $ & ' "$ ! ! !"$ . , ! $$"$ $$ ' -'$$ $$ $ ' ' $ $$ $'$ "$. $'" ! $$'& ' !& '& $ ! '& . .'$, ' "$ ! & $ &" $$ ". / ', ' "$ ! $

!! $ , &$ ! 6,7] ($'" ' ! !) 8] ($'" ' $

). ! 8] &$ !"$ $$ ! , $ $ ' "$ ! " 1. 2" "$ ! ' -'$$ F12 $ & 3$ ', $& '! $ '! 24 3 +' ! !"$ ! ' -'$$ 6, 7]. 5$ ' "$ !

F12 6,7] 1 6 z 1 + 6 1 2 0 2 ;2 1 2 2 0 z F = ; (;z) F 1 + + 0 1 02 21 1 1 + + ; 02 2 1 (;z);1 F 1 +1 0 16 z (1) 01 12 1 , ! $

-

89

]: "

/

# . ; # , !,

. -

S S

S@. A ", , , !

. 2 , ! , . 2

S T ! " . a e f 0 a0 e0 f 0 00 a 0 0 a 0 a0 00 00 a0 00 S: e 0 e 0 0 T : e0 a0 e0 00 00 : f 0 0 f 0 f 0 00 00 f 0 00 0 0 0 0 0 00 00 00 00 00 2 , S T

S 1 a = fa 0g S 1 e = fe 0g S 1 f = fa f 0g S 1 0 = f0g (1) 1 0 0 0 0 1 0 0 1 0 0 0 1 0 0 0 T a = fa 0 g T e = fe 0 g T f = fa f 0 g T 0 = f0 g: (2) 8 # '

S T , x x0 x 2 S , "

S T , (

) 1 '(xy) 2 h'(x) '(y)i ';1 ('(x)'(y)) 2 hx yi: (3) C

S T (1) (2)

. D , ' " " #

S ! #

T , # " #

S ! #

T , # E . ; , ' . 8 ' , , fe0 00g

fe 0g

S ,

T . F

, , . ; ,

. G , " '

S

T " , 1 ; $" $ ! , " ,'#

A S B T ( $ '(A) ';1 (B ) "$ ", "

$ T S $ $.

90

. .

/ x y 2 S 1 (3). +# ", "! ,

, " 0 x y z 2 S x = yz '(x) 2 h'(x) '(y)i1 0 2 T = ';1 () 2 h';1 ( ) ';1 ( )i1. 8

, !

. H

#

. I J (A) $ L(A)] $ ]

, #

# A. . A / a, J (fag) L(fag) , , J (a) L(a).

1.

S T : ( ) | $ ]

() ! A S

J (A) = J (A) $L(A) = L(A)] () !" x y 2 S hxi J (y) $hxi L(y)]

hxi J (hyi) $hxi L(hyi)].

( ). ( ) ) (). 2 | S T , A 2 2 Sub S , A 6= ?. ; A J (A), A J (A). 2 J (A) | , J (A) J (A). 2 # ;1 , ;1 J (A) J (A). J ,

S , # ! A, , / ,

J (A) = J (A). () ) ( ). 2 J (A) = J (A) A 2 Sub S , A 6= ?. H

I S . ; J (I ) = J (I ) = I , . . I T . K , ;1 K S K T . " () ) () () ) () # # . + , J - $L- ]

... J - $L- ].

#

. $ $ ]

S T % !% '. & ' " ' J $ L], . . !" a b 2 S

91

aJ b , '(a)J '(b) $aLb , '(a)L'(b)]: ) , ' " J - $L- ] %

J - $L- ] S J - $L- ] T.

L (. 24.1 24.2 $1, 2]

31.1 31.2 $2]),

S T " " ' # B / , ! , /

S

! # T , x 2 B

n h'(x)i = hxi '(xn ) = '(x)n . 5 # ' ", . F

J - , J - /

2. $ S | |

S T. & T + , + " S

" S

T. ) , ! (. . )

" . x

3. ! "# , %&#%

2

$ ], # xy = x $xy = y]. 2

, . 1. $ | $ ]

-

S T , ' | !% , % . & !" e f 2 ES e 6 f , '(e) 6 '(f ):

. - . H S T , e 6 f '(e) 6 '(f ). 2 e < f , . . ef = fe = e 6= f . 2 #, '(e) '(f ). H (4) . H , , '(f ) 2 J ('(e)). H # e < f e 2 J (f ), # 1 '(e) 2 J ('(f )). ; '(f ) 2 J ('(e)), , (4) / . - F = J (e)=I (e)

S . 8 # e f 2.38 $5] 0-

. 2 e < f , F # 0-

. 2 #, F ! "

B = ha bi, f " ab = f , ba 6 e. ; eJ f , f = set s t 2 S . 2 e < f , ! # s t 2 fSf . f = se et, f 6= et se, f = ef = fe, # e < f . 2# a = se b = et. D , a b # "

B , ab = f , ba = etse 6 e. H $7] (. # $2, 41.8]) ' B " , / e 2= B (5) '(e) < '(f ). M , ba < e. F , '(f ) " '(B ). J , ea 2= B: (6) n m n m F , ea = b a e = ef = eab = b a b 2 B , (5). 2 , ea /

. F : n m (ea)n = (ea)n+m : (7) G , ba < e ae = (aba)e = a(ba)e = aba = a, . . ae = a: (8) n n + m H (8) (7) ea = ea . F # a (8), an = an+m , # "

B . ; , ea / , / '(ea). '(ea) 2 h'(e) '(a)i. H '(e) '(f ),

: f'(e) '(f )g |

'(e) < '(f ). 2 f'(e) '(f )g |

, ,

,

(

). ; '(f ) | " '(B ) '(f ) = '(f )'(e), '(a) = '(a)'(f ) = '(a)'(f )'(e) = '(a)'(e), . . '(a) = '(a)'(e). ; h'(a) '(e)i = h'(a)i h'(e)i '(e)h'(a)i. ; '(ea) 2= h'(a)i h'(e)i, '(ea) 2 '(e)h'(a)i, . . '(ea) = '(e)'(a)n n. F , '(f )'(ea) = '(f )'(e)'(a)n = '(f )'(a)n = '(a)n = '(an ), . .

93

an 2 hf eai. 2 e < f , , hf eai = hf i heai. 8 an 2 heai f = anbn 2 eS , . . ef = f , # e < f . 2 '(f ) < '(e). ; '(f ) | " '(B ), '(e)'(a) = = '(a)'(e) = '(a). ; , h'(e) '(a)i = h'(e)i h'(a)i '(ea) 2 2 h'(a)i, . . ea 2 hai B , (6). 2

. 2 | . L , , e f 2 ES e < f '(e) < '(f ). H , ef = e e 2 L(f ), # () # 1 '(e) 2 L('(f )), . . '(e)'(f ) = '(e). . '(e) 1, g = ge = = gab = cgn ak b = cgn ak;1, . . g = cgn ak;1 2 Ca, (9). M , k = 1, . . ga = cgn a g = gab = cgn ab = cgn, . . g = cgn , , g 2 hc gni. c = g g;n 2 G, c 2 hgn ai, (9) , c 2 hgn i. M , g 2 hgn i gn # G. 2 r > 0. 2 #, / k = 1. 2 #, k > 1. 2 (10), : ga = = cgn;1(ga)ak;1 gnr = cgn;1 (cgn ak gnr )ak;1gnr = cgn;1cgn ak gnr;1(ga)ak;2 gnr = = cgn;1 cgn ak gnr;1 (cgn ak gnr )ak;2gnr . 2# , ga = cgn;1(cgn ak gnr;1 )k;2cgn ak gnr agnr. 2# d = cgn;1(cgn ak gnr;1 )k;2cgn . ; d 2 C ga = dak gnr agnr . F , ga = dak gnr;1(ga)gnr , ga = (dak gnr;1)m (ga)(gnr )m m. 2 jgj / g. 2 m = jgj ga = (dak gnr;1 )m ga = hgnr a, h = (dak gnr;1 )m;1 dak . 2 k > 1, , h 2 Ca. ga = hgnr a g = hgnr , h 2 hgi. J , h 2 Ca,

(9). A , k = 1 (10).

- (10) ga = cgn agnr : (11) n ; 1 nr n ; 1 m 2# m = jgj. (11) ga = cg (ga)g = (cg ) ga, . . ga = (cgn;1)m ga. F # bg;1, e = (cgn;1 )m . ; c 2 hgn ai, (9) c 2 hgn i. (11) , ga = gnl agnr ! l, a = gnl;1 agnr : (12) F #, n jgj / . 2 ! t, tn jgj gt 6= e. (12) a = (gnl;1 )ta(gnr )t = g;ta, g;t = e. 2

. G 2. 2 |

S T . G

95

G S . 8 , , G

. ; G !

/ . 2# hgi = ;1 h i. - , g 2 G. - J - Jg S J T Fg F , ! / J - . ; G Jg , G J . 2 F #

/ , # ",

h i F ( " 2 h i J ), , F #

, # 0-

. ; , F 0- , 0-

# . 2 1 Fg # # . D , Fg # 0- , 0-

. 2 e " G, h"i = hei. 2 2.54 $5] Fg "

B = ha bi " e, e = ab. 2# A = hg a bi #, A 1. ; h i, , ! / Sub T , , hgi ! / Sub S . 2 # hgi

,

g 2 G, , #

". M , ;1 h"i = hgn i n. 2 #, ga 2 hgn ai. H $7] B "

. 2# = '(a). 2 hgi \ hai = ?, ga 2= hgi hai ga 2 hg ain(hgi hai). M , hgai hg ain(h i hi), hgai h" in(h"i hi). ; , ga 2 hgn ai. H 1 A . , hgn i = hgi . 2

, 2. 8

. 1. , ! $ ]

S T !% ' + S n Gr S + T n Gr T, ! x 2 S

hxi = h'(x)i. $ !" x y 2 S n Gr S xJ y , '(x)J '(y) $xLy , '(x)L'(y)]: - , x 2 S n Gr S n | , xn 2= Gr S, '(xn ) = '(x)n + " : ) |

) S | ) ! x | !

, ! xk 2 Gr S, xk;1 2= Gr S , k 6= 4. . . x 2= Gr S, hxi ! / Sub S , . . hxi | ! / Sub T . 2 3.1 ) $1] $2] hxi = h i 2 T . H 2

96

. .

, 2= Gr T . > , # x 7! ". ; # 1. F # , x | / , '(xn ) = '(x)n 24.2 $1, 2], . # $2, 31.2]. . x | / , hxi\Gr S 6= ?. 2 k | , xk 2 Gr S . . k 6= 4, '(xn) = '(x)n $8]. 2 k = 4. ; $8] '(xn ) = '(x)n , '(x2 ) = '(x)3 '(x3) = '(x)2 . 2 #, # # ) ). A . , '(x2) = '(x)3 '(x3) = '(x)2 . 2 #, | . 8 , '(x)3 2 L('(x)2 ). 8 x2 2 L(x3 ), . . x2 = yx3 ! y 2 S . 2 e | hxi xm = e. ; x2 = yx3 = y x2 x = = ym x2 xm = ym x2e x2 = x2e, # x3 2= Gr S . 2

). 2 S |

| . ; '(x)3 2 J ('(x)2 ), x2 2 J (x3 ), . . x2 = yx3 z

! y z 2 S . H s , / z s , # z s = f . ; x2 = yx x2 z = (yx)s x2 z s = y1 x3f ! y1 2 S , , x2 = y1 x3 f , x2 f = x2 . 8 x3 f = x3 , . . x2 = y1 x3. L ! ",

, . 2. , ! $ ]

S T ! , !

|

.

x

4. , ( !

-

, #

. H $6, . 105]

AS-

. I S

/ "

S . 3. $ |

AS - S T. & T AS - , -

! " A S A " T + " " S

S=S T=T .

97

. L (., , $6, . 105]),

AS-

, / a b , a b , a2 bn ! n. 2 /

T . G 2 T ,

, . . 2 J (): (13) . 2 J (2), J () = J (2 ), : 2 J (2). 2 #, 2= J (2): (14) ; , , hi

, / hi ! / Sub T ;1 hi "

S , # C = ;1 hi = hai. 2 #, a 2= J (a2): (15) 2 # : a 2 J (a2 ). ; J (a) = J (a2 ), # 1 J () = J (ha2 i): (16) J , 2 ha2 i. F , 2= ha2 i, ha2 i hi, . . ha2 i J (2), (16) J () J (2 ), (14). F

C = ha2i, , C "

. 8 ,

D C J (C ) = J (D). H D = ;1 ha2 i, J (a) = J (C ) = J (;1 h2i), # 1 J () = J (2), # (14). 2# D = ;1 h i. (13) # 1 D J (C ). ; S AS-

,

, , D \ J (C 2 ) 6= ?. .! # 1, , h i\ J (C 2 ) 6= ?. 2 C 2 J (a2 ), (15) J (a2 ) J (2), h i \ J (2) 6= ?, . . 2 n ! n. , T AS-

. 2 A | S B = A. 2 #, B T . 2 2 B . J , T , . . # . ; ;1 hi ;1h i A, A ;1 h i \ J (;1 hi) 6= ?. 8 !

# 1 h i\ J () 6= ?. M , AC (x) AS-

, # ! / x, AC ( ) 6 AC (). 2 , AC () 6 AC ( ). M , AC () = AC ( ) B # C

T . 2 #

98

. .

;1 C , , # # S . ; , ;1 C A, , B = A, B = C . 2 #

# 2. 8

. + ,

#

, /

. 5 # A l S , A

S . I hhaii

, #

/ a

.

4. $ S |

, |

S

T. & T

, ! X S X T, Sub X

. . X | S, Sub X %

. / " S

T . . H 2 2

T . G X

S

#,

A l X , A l X: (17) 2 A l X . 8 M E S , X , N | # M X . P , M N | S , / M N l T .

2 #,

A l X , A M l S:

(18) 2 A l X , a 2 A M , s 2 S . J , sa 2 A M . . sa 2 M , . 2 sa 2= M . ; , a 2 N , # sa 2 N , . . sa 2 X . 8 , b 2 hhaii sb 2 X . 2 b | " hhaii, a = ba. sa = sba = (sb)a 2 A, A | X a 2 A. , sa 2 A M . 2 (A M ) l S , a 2 A, x 2 X . ; xa 2 X \(A M ) = A. J (18) . 2 (18) ,

(A M ) l T , 2 #,

A M l T:

99 (19)

B l X , B M l T: (20) 2 B l X , 2 B M , 2 T . J , 2 B M . .

2 M , . 2 2= M . ; , 2 N 2 L( ), a ;1 L( ) N , , 2 N , . . 2 X . . 2 M , L( ) M 2 M . M , 2 X , . . 2 B . 8 , 2 hh ii 2 X . 2 " | " hh ii, = " . = " = ( ") 2 B , B | X . , 2 B M .

8 " (20) #, (18). (19) (20) , A l X . H

S T # (17) . ; , X

. . X |

, !

, / # 31.1.2 $1], . # $2, 38.1.1]. . X |

, , X |

,

. . X |

, X |

. H 30.8 $1] (. # $2, 37.8]) X |

, . . X / . H

, # # . > ,

,

,

# 2.7 $1] $2],

, ! . , X |

.

, , Sub X / " . H

X

30.8 $1] Sub X " . 2

, # " . ; X |

T , # Y

T . J , ;1 Y

S T , , ;1 Y = X , . . X T . 2 # # 2 ,

S / !

T .

100 x

. .

5.

+ ,

S

T " ' # S T , x y 2 S '(xy) 2 f'(x)'(y) '(y)'(x)g: (21) + ' , x y 2 S f'(xy) '(yx)g = f'(x)'(y) '(y)'(x)g: (22) M! , ! . + , !

ha b j a2 = a b2 = b ab = ai

! . > , , # # . H / # ! # .

5. -+

%

. , ! S T !

, !" , !

S T. F # .

2. $ ' | !% , %

S T, X Y | S, '(X ) '(Y ) | T, X 6 Y '(X ) 6 '(Y ). . ' X

$

], !" x 2 X, y 2 Y '(xy) = '(x)'(y), '(yx) = '(y)'(x) $'(xy) = '(y)'(x), '(yx) = '(x)'(y)]. . F , ' X ,

#, '(xy) = '(x)'(y). H . P , X < Y . M '(X ) < '(Y ). ; x xy 2 X , '(x) = = '(x xy) = '(x)'(xy), . . '(x) = '(x)'(xy): (23) 2 #, (23) '(xy) = '(x) '(xy) = '(x)'(y):

(24)

101

; '(xy) 2 h'(x) '(y)i '(x) 2 '(X ), '(y) 2 '(Y ), '(X ) < '(Y ),

,

, #

/ '(x) '(y), ( ) / '(x) '(y) '(x)'(y) '(y)'(x) '(x)'(y)'(x) '(y)'(x)'(y) , '(xy) 2 f'(x) '(x)'(y) '(y)'(x) '(x)'(y)'(x) '(y)'(x)'(y)g: ; '(Y ) |

'(X ) < '(Y ), '(x)'(y)'(x) = '(x): (25) ; , '(xy) '(x) '(y) '(y), '(xy) = '(x)'(y),

'(x), , # '(xy) = w('(x) '(y))'(x)

'(x) (23) (25), '(xy) = '(x). J , (24) '(x)'(y) = '(x). H , '(xy) = '(x) xy = x, . . hx yi = fx y yxg. . '(x)'(y) 6= '(x), '(X ) < '(Y ) '(x)'(y) = '(yx), . . '(x)'(y) = '(yx x) = '(yx)'(x) '(x)'(y) = '(yx)'(x), '(x)'(y) = '(x)'(y)'(x) = '(x). 2

, '(xy) = '(x)'(y). F

, ' X , '(xy) = '(y)'(x) '(yx) = '(x)'(y).

3. $ ' | !% , %

S T , X Y | S, Z | " " S, '(X ) '(Y ) | T, '(Z ) | " " T. . ' Z

$

], !" x 2 X, y 2 Y '(xy) = '(x)'(y) '(yx) = '(y)'(x) $'(xy) = '(y)'(x) '(yx) = '(x)'(y)]. . 2 #, ' Z , #, '(xy) = '(x)'(y). H

. 2 2, '(xy) = '(x)'(xy) = = '(x)'(xy)'(y), . . '(xy) = '(x)'(xy)'(y). ; '(xy) 2 h'(x) '(y)i,

. F

, ' Z , '(xy) = '(y)'(x) '(yx) = '(x)'(y). F # 5 . 2 S |

, ' | # , "!

S

T . H 4 X

S '(X )

T , 'jX

102

. .

'

S . 2 2 3, # , ' . - # 5. F ! # .

4. )

S T %

S T. F # . 2 ' | " , "!

S T . H # 2 # S T . M 30.8 $1] ' S . ; , (22) # , x y # S , # X Y . H S T , # x y, (21). - ! . 2 X Y . -

#, X < Y ( Y < X

). ; # 2 '(X ) < '(Y ). F , ' X . ; '(xy) = '(x)'(y) 2. K 2 , ' X , '(xy) = '(y)'(x). 2 X Y . 2 Z | X Y

S . 2 #, ' Z . ;

# 2 '(X )'(Y ) '(Z ), 3 '(xy) = '(x)'(y). K 3 , '

X , '(xy) = '(y)'(x). F # 5 . ; , 5 . K # >. +. Q " .

*

1] . ., . . . ! 1. | #: %&- ( . - , 1990. ! 2. | #: %&- ( . - , 1991. 2] Shevrin L. N., Ovsyannikov A. J. Semigroups and their subsemigroup lattices. | Dordrecht: Kluwer Academic Publishers, 1996.

103

3] . . % ! . &/0&/ // 2! /3 0 4 5. #5. &. | 6 , 1993. | #. 242. 4] Ovsyannikov A. J. On ideal lattice isomorphisms of semigroups // Colloquium on Semigroups. Szeged, 15{19 August 1994. Abstracts. | P. 28. 5] 600 ., ;. 5 . . 2. 1. | 0 gk (x) d < 1, gk (x) 1 P

k=1

*. 6, . 349{350].

x

1. (C,1)-

% ( |

) # & % # # ,

f(k g1 (, ) (k , k=1 | 2

) 1 S (0 = ?, (k (k+1 ! k 2 N, (k = (. k=1

.R %!c(!) | ) ! 2 &! (. % % 1

fsk gk=1 , $ sk = c(!)e d(!), $ $ !

k

( ),

R c(!)e! d(!).

-

1 $ % % fn g1 n=1, $ n = n (s1 + : : : + sn ) | % % (C,1)- ( 2 ) , ) 1 !) ! fsk g1 k=1.

107

. % % fsk g1k=11 !

(C,1)- S, % % fngn=1 ! S n ! 1. % fe! g!2 | ! !, c(!) | ) ! 2 &! (. " 1. n f n g1 n=1 +1 1 < q 6 n , 1 2 Z X q 2 ksn ; n k 6 q2 ; 1 jc(!)j2 d(!): (1) n=1 # . ; ! n , ) Z n Z X 1 ! sn ; n = c(!)e d(!) ; n c(!)e! d(!) = k=1 k n Z Z n n X k X X = c(!)e! d(!) ; n1 c(!)e! d(!) = k=1 m=1 m n m;1 k=1 k n k;1 Z Z n n X X c(!)e! d(!) = = c(!)e! d(!) ; n1 (n + 1 ; k) k=1 k n k;1 k=1 kn k;1 Z n X = n1 (k ; 1) c(!)e! d(!) k=1 kn k;1 X 2 Z n c(!)e! d(!) 6 ksn ; n k2 = 12 (k ; 1) n k=1 k n k;1 Z n X 6 12 (k ; 1)2 jc(!)j2 d(!) n k=1 k n k;1

1. $ Z n 1 1 X X X 1 2 2 ksn ; n k 6 2 (k ; 1) n=1

n=1 n k=1

6

X 1

jc(!)j2 d(!) 6

k n k;1

(k ; 1)2

k=1

Z k n k; 1

jc(!)j2 d(!)

; ! n+1 > q n 1 X 1 1 X 1 = q2 : ;2m = 1 q 6 2 2 k2 1 ; q12 k2(q2 ; 1) n >k n k m=0

X 1 2: n >k n

108

. .

= . 1 1 2 X X ksn ; n k2 6 q2q; 1 n=1

Z

k=1 k n k;1

jc(!)j2 d(!) =

q2 Z jc(!)j2 d(!): q2 ; 1

.

" 2. 1 X

n=1

nkn+1 ; nk2 6

1 Z jc(!)j2 d(!): 2

(2)

# . ; ! sk n sk =

Z

c(!)e! d(!) =

k

k X

Z

m=1 m n m;1

c(!)e! d(!)

n X n+1 ; n = n +1 1 sn+1 ; n(n1+ 1) sk k=1 Z Z n n k n X XX X sk = c(!)e! d(!) = (n + 1 ; k)

c(!)e! d(!)

k=1 m=1 m n m;1 k=1 k n k;1 Z Z n n n X X X 1 1 1 ! c(!)e d(!) ; n(n+1) k c(!)e! d(!) n(n+1) k=1 sk = n k=1 k=1 k n k;1 k n k;1 k=1

+

n 1 Z X 1 n+1 ; n = n + 1 sn+1 ; n c(!)e! d(!) ; k=1 kn k;1 Z Z n n X X c(!)e! d(!) = n1 c(!)e! d(!) + ; n(n1+ 1) k k=1 kn k;1 k=1 k n k;1 Z n nX +1 Z X + n(n1+ 1) k c(!)e! d(!) + n +1 1 c(!)e! d(!) = k=1 kn k;1 k=1 k n k;1 X Z n c(!)e! d(!)+ = n +1 1 ; n1 k=1 k n k;1 Z Z n X 1 1 ! + n(n + 1) k c(!)e! d(!) = c(!)e d(!) + n + 1 k=1 kn k;1

n n n;1

109

= n +1 1

Z n n n;1

= n(n1+ 1)

Z n X c(!)e! d(!) + n(n1+ 1) (k ; 1)

nX +1

Z

k=1

kn k;1

(k ; 1)

k=1

c(!)e! d(!) =

k n k;1

c(!)e! d(!):

kn+1 ; nk2 6 n2(n1+ 1)2

nX +1

Z

k=1

k n k;1

(k ; 1)2

jc(!)j2 d(!)

1. ; %! )

. & , Z 1 1 nX +1 X X nkn+1 ; nk2 6 n2 (nn+ 1)2 (k ; 1)2 jc(!)j2 d(!) 6 n=1 n=1 k=1 k n k ;1 Z n +1 1 X X jc(!)j2 d(!) = 6 n13 (k ; 1)2 n=1 k=1 k n k;1 Z nX +1 X = (k ; 1)2 jc(!)j2 d(!) m13 6 k=1 m>k k n k;1 Z Z 1 X 6 12 jc(!)j2 d(!) = 12 jc(!)j2 d(!) k=1

k n k;1

X 1 Z1 dt 1 3< 3 = 2(k ; 1)2 : m t m>k k;1

. % p q | !

: p > q, % .4! % % f n g1 n=1 !

(3) 1 < q 6 n +1 6 p: n

% % H = L2 (X) |

$ ) #

& % # # , 2 &! c(!) , ) Z jc(!)j2 d(!) < 1: (4)

3. ! (4). " ! ! (C,1) X , , ! # f n g,

110

. .

# (3) ! fsn (x)g .

# .

. % % % fn(x)g ! ) .

X. ; ! & (1), 1 Z 1 X X 2 jsn (x) ; n (x)j d = ksn (x) ; n (x)k2 6 n=1 X n=1 2 Z 6 q2q; 1 jc(!)j2 d(!) < 1: 1 P !! 2, ) % ) . ! jsn (x) ; n=1 ; n (x)j2, , ) ) . X nlim j s (x) ; n (x)j = 0, n !1 . . % % fsn (x)g ! ) .. . % fsn (x)g ! ) .. 3 (1) % % fn (x)g ! ) . X. ?

% k (x) ; n (x) ! n < k < n+1 : k X k (x) ; n (x) = (j (x) ; j ;1(x))

j =n+1

X 2 k jk(x) ; n (x)j2 = (j (x) ; j ;1(x)) 6 j =n +1 k p k 2 X X 1 pj j j(j (x) ; j ;1(x))j2 6 6

6

j =n +1 X n+1

X n+1

j =n +1

1 j jj (x) ; j ;1(x)j2 j j =n +1 j =n +1

@{A ! ${B &. D) !, ) n+1 6 p n , ) pn X pn n+1 16 X 1 6 Z dt = lnp: t j =n +1 j j =n +1 j n

= .

jk(x) ; n (x)j2 6 ln p

X n+1 j =n +1

j jj (x) ; j ;1(x)j2:

(5)

111

; %! & (2), ) 2 % ) . X 1 P ! njn(x) ; n;1(x)j2, ), (5) ! . ) n=1 . X, + % % fn(x)g1 k=1 !4 # ! ) . X. . 1 % 1. P jckj2 < 1, f'k(x)g1k=1 | k=1 1 P $ L2. " ! !% k 'k (x) k=1 X (C,1), , ! # f ng, # (3), fsn (x)g . # . 3 ! 3 (= N, (k = f1 2 : : : kg, 1 R P (!) 1, jc(!)j2 d(!) = jck j2. k=1

% 2 ( . &4, . 127]).

1 P

jck j2 < 1, f'k (x)g1 k=1 | k=1 1 P

$ L2 . " ! !% k 'k (x) k=1 X (C,1), , ! # f n g, # (3), fsn (x)g .

# . E

! 1, ) !, )

! !. ! ) )

& % # #, ) 2. x

2. (C )-

. F An = ;n+n ) n-# +22& %-

1 P

$ ! Ant = (1;t1)1+ ( 6= ;1 ;2 : : :), An | ) F ! . n=0 ; $ ! % +22& ! ( . n n P P 4, . 75]) , ) An;;k1 = An G An++1 = Ak An;k G An = n+ An;1 G k=0 k=1 An;1 = + n An . , ) An n . ! . ; ! An 1 n n X 1 X X log An = log 1 + k = k + O k2 : k=1 k=1 k=1

+, )! ) C !

. H# ,

112

. .

X 1 j logAn ; log nj 6 C + o(1) + O k12 < M k=1

$ M | . ! !

!G . 4 %

M1 M2 , ) n > 1 M1 < Ann < M2 : (6) = . ! An;;k1 =An ) & (n ; M + 1);1 1 An;;k1 = O = O (7) An n n :

. n() = sA(nn) , $ s(n) | ) ! ! , -

. + , . I 1 > 2, (C 1)

, ) (C 2), . . (C 2)- # $ (C 1). 3 ) n()(x) ! ) ) $ % Z n X 1 ( ) n (x) = A An;k c(!)e! (x) d(!): n k=1

k n k;1

E %, ) s(n) (x) ! ! ! n- +22&

$ ! 1 P sn (x)tn 1 X s(n) (x)tn = n=0 (1 ; t) : n=0

% > ;1, >10. $ s(n+) (x) ! ! ! n- +22& -

P sn(x)tn

! ! (1;t) (1;1t) , , n n P P ! , s(n+) (x) = An;;1k s(n) (x) = An;;1k Ak n()(x), n=0

k=0 k=0 n n X X s(n++1) (x) = s(k) (x)An;k n(+) (x) = 1+ An;;1k Akk() (x): An k=0 k=1 n ( ) 1 P = n()(x) = n+1 jk (x) ; k(;1)(x)j2. k=1 " 3. (4) > 21 lim () (x) = 0: n!1 n

(8)

113

# . ;

k() (x) ; k(;1)(x) = Z Z k k X X = A1 Ak;j c(!)e! (x) d(!) ; 1;1 Ak;;j1 c(!)e! (x) d(!) = A k j =1 k j =1 j n j;1 j n j ;1 Z k X = 1;1 Ak;j Ak ;1 ; Ak;;j1 Ak] c(!)e! (x) d(!) = Ak Ak j =1 j n j;1

Z k X ;1A;1 = = 1;1 A c(!)e! (x) d(!) +k ; j Ak;;j1Ak ;1 ; +k k ;j k Ak Ak j =1 n j j ;1 Z k X 1 ! = ;1 c(!)e (x) d(!) ; j Ak;;j1Ak;1 = Ak Ak j =1 j n j;1 Z k X 1 ;1 = A (;j)Ak;j c(!)e! (x) d(!) k j =1 j n j;1

% , 1 Z Z k X jk()(x) ; k(;1)(x)j2 d 6 2 (A1 )2 j 2 (Ak;;j1 )2 k j =1

X

Z X

jc(!)j2 d(!)

j n j;1

2(n )(x) d 6

Z 2n k X 1 X 1 ;1 2 jc(!)j2 d(!) = ( )2 (2n + 1) k=1 (Ak )2 j =1 j Ak;j j n j ;1 n Z 2 2n A;1 !2 X 2 X 1 k;j : 2 = 2(2n + 1) j jc(!)j d(!) A k j =1 k=j j n j;1

D) ! ! (6) (7), ) ;1 !2 2j A;1 !2 2n A;1 !2 X 1 X X Ak;j k;j k ;j 6 + 6 A A A

k k=2j +1 j 1 X C1 A 2 + X M2 k;1 2 6 (A1 )2 k j k=0 j k=2j +1 M1 k ) . ) %, > 12 , % ) k=j

k

k=j

k

114

. .

1 1 1 1 (j + 1) C1 M j 2 + M2 2 X M12 j 2 j 2 M1 k=2j +1 k2 6 2 2j 1 1 = C2 : 2 2 2 2 ; 2 6 M 2 j 2 C1 M2 j + M M1 2j j 1 , Z Z 2n X C 2 ( ) jc(!)j2 d(!) 2n (x) d 6 22n j j =1 X

j n j;1

1 Z X

Z 1 2n X X 1 ( ) 2n (x) d 6 C3 2n j n=1 X n=1 j =1 j n j;1 Z 1 X 2 6 C3 j j j =1

jc(!)j2 d(!) 6

jc(!)j2 d(!) = 2C3

Z

jc(!)j2 d(!) < 1:

j n j;1

;, nlim (n ) (x) = 0 ) .. !1 2 J! 2n < k 6 2n+1 : 1 ) (x) = 2(n+1 n +1 2 +1

> 21 k +1 1 > 21 k +1 1 k() (x)

n+1 2X

m=1 k X

n+1 2X

m=1

jm()(x) ; m(;1)(x)j2 >

jm()(x) ; m(;1)(x)j2 > jm()(x) ; m(;1)(x)j2 = 12 k() (x)

m=1 ( ) 22n+1 (x),

0 6 6

% ) . klim () (x) = 0, !1 k ) %. .

4. ! (4). R c(!)e! (x) d(!) E f(x) (C ) > 21 , E n 1X jf(x) ; k(;1) (x)j2 = 0 (9) lim n!1 n k=1 n 1X lim jf(x) ; k(;1) (x)j = 0: (10) n!1 n k=1 # . J ) (9). ;

115

n X k=1

jf(x) ; k(;1) (x)j2 =

n X

jf(x) ; k() (x) + k() (x) ; k(;1) (x)j2 6

k=1 n X

62

k=1

jf(x) ; k() (x)j2 + 2

n X k=1

jk() (x) ; k(;1) (x)j2:

!! 3 ! o(n), !! ! o(n) !. H

(9). n jf(x) ; (;1) (x)j X k 6 n k=1 X 12 X 1 n n n X 2 2 6 jf(x) ; k(;1) (x)j2 n12 = n1 jf(x) ; k(;1) (x)j k=1 k=1 k=1

@{A ! ${B &. = . n jf(x) ; (;1) (x)j2 21 n jf(x) ; (;1) (x)j X X k k 6 nlim = 0: lim !1 n!1 n n k=1 k=1 %..

5. (4) ! (C 1) & { !

E . # . R 1. % c(!)e! (x) d(!) ! (C,1) 2 & f(x), ) # ) . E. 3 % ! K ! p pX ;1 X k bk t = (b1 + : : : + bk )(tk ; tk+1 ) + (b1 + : : : + bp )tp = k=1 k=1 pX ;1 = (1 ; t) (b1 + : : : + bk )tk + (b1 + : : : + bp )tp : k=1

Z pX ;1 ! k c(!)e (x) d(!) = (1 ; t) sk (x)t + c(!)e! (x) d(!)tp = k=1 k n k;1 k=1 p p;2 X k 2 = (1 ; t) kk (x)t + (p ; 1)p;1 (x)tp;1 + sp (x)tp : k=1 ; jsp (x)j 6 pC1, jp;1(x)j 6 C2, Z p X k Ft(x) = plim t c(!)e! (x) d(!) = !1 k=1 k n k;1 p X

tk

Z

116

. .

pX ;2 2 k p ; 1 p = plim (1 ; t) kk (x)t + (p ; 1)p;1(x)t + sp (x)t = !1 k=1 1 X = (1 ; t)2 kk (x)tk : k=1

1 P @ $, ! t < 1 (1;1t)2 = ktk;1, k=1 1 1 X X jFt(x) ; s(x)j = (1 ; t)2 kk (x)tk ; (1 ; t)2 ks(x)tk;1 = k=1 k=1 1 X = (1 ; t)2 ktk;1(k (x)t ; s(x))] = k=1 N 1 X X 2 k ;1 2 k ;1 (kt (k (x)t ; s(x))): = (1 ; t) (kt (k (x)t ; s(x))) + (1 ; t)

k=1

k=N +1

3 N, 1 > t0 > 0, 1 > t1 > 0 ! % $ " > 0 , ) jk(x) ; s(x)j < "4 ! k > N, jk (x)t ; s(x)j < "2 ! k > N, t > t0 . N N P P (1 ; t)2 ktk;1jk (x)t ; s(x)j 6 (1 ; t)2 ktk;1C3 < (1 ; t)2 C3N 2 < "2 k=1 k=1 t > t1 (t1 ) 1). $ ! k > N, t > maxft0 t1g 1 1 X X ktk+1 2" < 2" + 2" (1 ; t)2 ktk+1 = 2" + 2" = ": jFt(x) ; s(x)j 6 2" + (1 ; t)2 k=N +1 k=1

* % , $ K !{

. R 2. % c(!)e! (x) d(!) ! K !{

) R . EG jc(!)j2 d(!) < 1. 1 P $ ! (n (x) ; n;1(x)) ) . E n=1 1 P . K + ) %. ! njn(x) ; n;1(x)j2 ( 1 n=1 P (3)) % ! (n(x) ; n;1(x)). * % , + ! n=1 R

! ) . E, $ c(!)e! (x) d(!) !

E ) . (C,1). %..

6. ! (4). R c(!)e! (x) d(!)

! & { E > 0, E (C ).

117

# . * ) , )

n 1X (r) (x)j2 = 0 r > ; 1 lim j k n!1 n 2 k=1

1

! " > 0 nlim n(r+ 2 +") (x) = 0. !1 3 , (6) , ) n n X X 1 1 1 s(nr+ 2 +") (x) = s(kr) (x)An;;2 k+" = s(kr) (x)Ark A;n;2 k+" = r+ 12 = ; 12 +": k=1

k=1

+ 1 n n; X (r+ 21 +") X 1 2 sn (x) 6 jk(r) (x)j2 Ark An;;2 k+" 2 6 k=1 k=1 v v v u u u n n n p X X u uX u ( r ) ;1+2" 2 r 2 t t jk (x)j K Ak An;k = K t jk(r) (x)j2 A2nr;1+2"+1 = 6 k=1 k=1 k=1 v u n p uX = K t jk(r) (x)j2 A2nr+2" = o(n1=2 )O(nr+" ) k=1

%. ; # 6, 5, $ ! (C,1) ) . E, ) 4 = 1 (9), . . n (0) 1 P jk (x) ; f(x)j2 = 0. ) . E nlim !1 n k=1

(0) ) . ! ; R? %! k (x) ; f(x) R

% c(!)e (x) d(!) ; f(x) + c(!)e! (x) d(!) + : : : % 1n 0 2n 1 ,

) % , r = 0. ), ) h ( 1 +") i 2 (x) ; f(x) = 0 lim n n!1 ) . E. ; %! 4 4 = 12 + ", ) .4# %: n 1 1X n(; 2 +") (x) ; f(x)2 = 0 lim n!1 n k=1 ) . E. = . 4%.

# .)%, ) lim (2")(x) ; f(x)] = 0 n!1 n ) . E.

118

. .

A ! % " = 2 , ) % . .

7. (4) R ! (C ) > 0 c(!)e! (x) d(!) ! % ! & { .

# .

1. J % % (C,1) (C ) ! 0 < < 1. ; (C ) (C,1)- %G 5

% K !{

G 6 (C )- %. 2. J % % (C,1) (C ) ! 1 < < 1. ; (C,1) (C )- %G % K !{

, % .$ (C )G 5 % (C,1). %.. % 3. " ! $ L2 ! (C ) > 0 ! % ! & { .

# . 3 ! 7 (= N, (k = f1 2 : : : kg, 1

R P (!) 1, jc(!)j2 d(!) = jck j2. k=1

% 4 ( . &5, . 219]). " ! $ L2 !

(C ) > 0 ! % ! & { .

# . D

4 $

! ) !,

$ 2. K $ 2

. . ,

# .

" # 1] . . , , " # # // % . - &. '. ()*. *. | ,*- -- :

/-* 0 0-%, 1996. |

3. 117{118. 2] . . *#* , // ' . . 3*. ) "# . * . | ' : '89, 1997. | 3. 105. 3] . . " // ' %* -. 3 % ., % . | 1997. | = 5.

4] ? @ 8. *. | %.:

119 / *

# , 1963. 5] 0@ 3., A # 8. *. | %.: 8/B%, 1958. 6] 0* ?. &., B 3. '. C ) "# ) " . | %.: &, 1989.

' ( ) 1997 .

SV- . . . . . 512.552

: , , V- , , , , , .

! " # $ SV- . % & SV- Soc (R) .

Abstract

V. N. Silaev, On right SV-rings, Fundamentalnaya i prikladnaya matematika,

vol. 7 (2001), no. 1, pp. 121{129.

In this paper we investigate the worst cases of SV-ring structure. We give two constructions of SV-rings with strong restriction on all Soc (R) of Loewy chain.

. R, x 2 R y 2 R, xyx = x. ! " #$ % & 1936 . ! + , ,, " , " , +, +$ -. . % /6]. 1 " , V- , , " 2. !% /7] 1964 . R, $ " R- . 5" " , ,. 6 +, " () "$ 9. -, " , " "+ , :. ; /5]. # ". 9+ = "+ " R- M, " " Soc (M) "

" : "$ Soc0 (M) = 0 $ " Soc +1 (M) " " Soc +1 (M)= Soc (M) = Soc(M= Soc (M)) (Soc(M) | M)A | " ,

, 2001, 7, - 1, . 121{129. c 2001 !, "# $% &

122

. .

S

"$ Soc (M) = Soc (M). 5 C < 6 jM j, Soc (M) = Soc+1 (M). D" 0 = Soc0(M) Soc1 (M) = Soc(M) Soc2 (M) : : : Soc (M) + ( ) M. E M + , Soc (M) = MA + M. - R + , RR ", " " , " ". F, "" % - , Soc (RR ) = Soc (R R), " " Soc (R) . 1C "+ " 10{15 " SV- , " " " V- , , + , " + . 6 , " /1,3,4], | /8]. 9 #$. 2 /2], $ "= , "+ " " : 1) " u- " SV- + 1A 2) " " SV- + 2, V- . -$ + , " , R , + % R= Soc (R). 2 + ", % J",K, "= " . 9 , " , " , SV- "+ +1 ( " $ " , . . " , ), < % R= Soc (R) " ( " +1 % - R= Soc (R)

"" " 9={L " "+ " , , " $ " ). 1 , " " . L " + , + %+ - , , "% L. 9. E,= %+ - , , "% #. 9. 5 + " + "+ $.

&C " " % " + #$. 2 /2]. M = , C +. 1 X | "+ $, D | . 5 + + CFMX (D) , X X D

SV-

123

$ . QC +, + " UD = D(X ) += +%+ CFMX (D) ' End(UD )A

, Soc(CFMX (D)) + , , . 2 $ D " , CFMX (D). 6 " , , + $ 8 6 " To Q CFMX (D) ", X, : 1) Q ' CFMX (D)A 2) Q Q 8 : < 6 A 3) Q \ Soc(Q ) = 0 8 : < 6 A 4) D Q 8: 6 A 5) Soc(Q ) Soc(Q ) Soc(Q ), = min( ). 5 "

" """ L Q1 : L0 = 0A L +1 =SL + Soc(Q +1 ) 8 < A L = L " 6 , < R = L + D " SV- + 1. 9 D " C $ " Q1 , "= ". 5" + . 1. # "+ > 0 " SV- R + 1, < % - R= Soc (R) " . . M " + 2. &"

+ /2] + "= " ". (2, lemma 4.1]). | X | , jX j = @ . < P X , 1) jY j = @ 8Y 2 P ! 2) 6 , P P . (2, proposition 4.2]). > 0 X | jX j = @ , D | Q = CFMX (D).

6 Q " # Q , $ Q 1) Q Q 8 6 6 ! 2) Q \ Soc(Q ) = 0 8 < 6 ! 3) D Q 8 6 . . # 6 " P | + X + . # $ Y 2 P

fY : @ ! Y + xYi = fY (i) $ i < @ .

124

. .

# $ x 2 X 9! Y (x) 2 P , i (x) < @ , x = xY (x)i (x) . L, < 6 , $ Z 2 P + @ $, "$, P , " S

gZ : @ ! P , Z = fgZ (j) j j < @ g, + YZj = gZ (j) 8j < @ . # Y 2 P 9! Z (Y ) 2 P , j (Y ) < @ , Y = YZ (Y )j (Y ) . # $ 6 " $: ' : CFMP (D) ! Q: (' (A)xy = i (x)i (y) AY (x)Y (y) A ' : CFMP (D) ! CFMP (D): (' (B)Y Y = j (Y )j (Y ) BZ (Y )Z (Y ) : : ", ' ' | %+ 8 < 6 ' = ' ' . # $ 6 Q = Im(' ). 1) 5 jP j = @ = jX j, Q ' Q. 2) 6+ ' = ' ' , Q Q 8 < 6 . 3) X 0 6= A 2 CFMP (D), , 9Z Z 2 P , AZZ 6= 0. F Y , Z = F Y , ' (A) 5 Z = Zj Z j YZj YZ j = AZZ 6= 0 8j < @ . j< j< F, ' (A) $ @ , . 1 Im(' ) \ Soc(CFMP (D)) = 0, Q \ Soc(Q ) = 0. Y, (Soc(Q )) 6 , """ Q Soc(Q ) \ Soc(Q ) = 0 " 6= A Soc(Q ) Soc(Q ) Soc(Q ) = minf g: # $ 6 " "

""" L Q1 : L0 = 0A L +1 =SL + Soc(Q +1 ) 8 < A L = L $ " 6 . < - R = L + D " SV- + 1. 9 D $ " Q1 , "= ". M " , +. !. 1 | , X | $ jX j = @0 . !" " DEMX (D) CFMX (D) = Q +: 0

0

0

0

0

0

@

0

0

0

@

DEMX (D) = = fA 2 Q j 9n = n(A) > 1 8i j : i > n j > 1 Aij = Ai+1 j +1 g: 5 DEMX (D) Soc(Q) Soc(DEMX (D)) = Soc(Q). X + A B 2 DEMX (D) 0

0

SV-

(

(

125

j = i + 1 B = 1 i = j + 1 Aij = 01 A ij 0 , A B = 1, B A 6= 1. - , DEMX (D)= Soc(Q) = D(x) | , " x, x = A mod Soc(Q), x 1 = B mod Soc(Q). [ "

2, P+1 , jP+1j = @0 . 9= + "$ + . !" " R( + 1) +: R( + 1) = L + '+1 (DEMP+1 (D)) jP+1 j = @0 : 5, '+1 (DEMX (D)) Q+1 Q 8 < + 1 | " " , Q Q 8 6 6 + 1, R( + 1) | " Q1 (L ) 6 L+1 = L + Soc(Q+1 ) L+2 = R( + 1) | + " R( + 1). # 6 L \ Q +1 = = 0, L | Q +1 + L . 5 L Q +1 + L '+1 (DEMX (D)) Q +1 , ", R( + 1)=L " (Q +1 +L )=L = Q +1 Soc(Q +1) = L +1 =L R( + 1)=L , , L +1 =L = Soc(R( + 1)=L ). #, R( + 1)=L+1 = '+1 (DEMX (D))= Soc(Q+1 ) = D(x) "", " (L ) 6+2 | " R( + 1), " ", + 2. # $, R( + 1)=L ". # = + 1 + 2 ". 1 6 . M % , $ X , a b 2 CFMX (D) + a Soc(CFMX (D)) b = 0 , a = 0 b = 0. # , b 6= 0 ) Im(b) 6= 0, a 6= 0 ) Ker(a) 6= UD = D(X ) , " x 2 Soc(CFMX (D)), " - + Im(b) + UD n Ker(a) , a x b 6= 0. 5" , " R( + 1)=L (Q +1 + L )=L = Q +1, x (R( + 1)=L ) y = 0 = x Soc(Q +1 ) y = 0, " + x = 0 y = 0, R( + 1)=L ". # $, fL j < + 1g | $ , R( + 1), $, L+1 . 1 I | R( + 1), L+1 6 I. = minf 6 + 1 j L 6 I g, , + 1 < + 1 L I. - R=L ", ""$ L 6= I, 0 6= (I=L ) (L+1 =L ) = (I \ L+1 )=L . M , Soc(R=L ) = L+1 =L | R=L , , L+1 I | ". # $ " , R( + 1) " V- . 9+ = " " R( + 1)- U. 5 UR(+1) = (R( + 1)=M)R(+1) " M R( + 1). X 0

0

0

0

0

0

0

;

0

126

. .

P = R( + 1) M = fr 2 R( + 1) j R( + 1) r M g $ L+1 , P = L+1 , L+1 | . X $ P 6 L+1 , " + P = L < + 1. 5 +, U " R( + 1)=L - ( " " P = Ann(U)). 1 U " R( + 1)=L - . 1 /2, theorem 2.5], R " " "+Q Q , Q = End(U )D , U | " D - ", ; L (D ) ; | Soc(Q ) R, " " R- . ; 9 C R( + 1)=L (

R( + 1)=L Q +1 ), " U R( + 1)=L - . & UR(+1) " + /6]. (6, lemma 6.17]). ': R ! S | $ #, A |

% S - . & R S | , A

% R- . $ (6, corollary 1.13]). ' # R , ( ) R- . 5 +, " R( + 1) | " SV- + 2 " (L ) 6+2 , R( + 1)=L " 6 + 1. 5" " | " . 9+ = D "+ " F M R = R( + 1) + F:

2

2

2

2. 5 " SV- R n " 0 = Soc0 (R) Soc1(R) : : : Socn (R) % - R= Socm (R) m, 0 6 m 6 n;2, " .

. 3. X | " , R | # . ( -

$ h: CFMX (R) ! CFMX (CFMX (R)): &. X | =, "

X $ P = f(i j) j i j 2 Z 1 6 i < 1 1 6 j < 1g. !+ + p1 p2 : P ! N "

" " . 5 p1 (P) p2(P ) $ = , + " +%+ h : CFMP (R) ! CFMp1 (P ) (CFMp2 (P ) (R)): 1 A 2 CFMP (R), " $ h

h (A) = a 2 CFMp1 (P ) (CFMp2 (P ) (R)) ai1 i2 2 CFMp2 (P ) (R) (ai1 i2 )j1 j2 = A(i1 j1 )(i2j2 ) . !, h (1) = 1, h (A + B) = h (A) + h (B). h (AB). X (h (AB)i1 i2 )j1 j2 = (AB)(i1 j1 )(i2 j2 ) = A(i1 j1 )(kl) B(kl)(i2 j2 ) =

0

0

0

0

0

0

0

0

0

= =

(kl) : B(kl)(i2 j2 ) 6=0

X

(h (A)i1 k )j1 l (h (B)ki2 )lj2 = 0

kl : (h0 (B)ki2 )lj2 6=0

X

0

X

(h (A)i1 k )j1 l (h (B)ki2 )lj2 = ((h (A)h (B))i1 i2 )j1 j2 0

l: k: h (B)ki2 6=0 (h (B)ki2 )lj2 6=0

0

0

0

0

0

, h (AB) = h (A)h (B). 1 h | %+ . 9+ = A 6= 0. 9 (i1 j1) (i2 j2 ): A(i1 j1)(i2 j2 ) 6= 0, (h (A)i1 i2 )j1 j2 6= 0, h (A) 6= 0. 1 h . 1 a 2 CFMp1 (P ) (CFMp2 (P ) (R)). 5 ((h ) 1 (a))(i1 j1 )(i2 j2 ) =(ai1 i2 )j1 j2 . 1 h , +. 1 F | ". # n > 1 + + Qn CFM | X (CFM{zX (: : :CFMX }( F) : : :)): 0

0

0

0

0

0

0

0 ;

0

n

128

. .

9 Q0 + = F. $ 4. n $ hnn+1 : Qn ! Qn+1 : . Qn = CFMX (Qn 1), Qn+1 = CFMX (CFMX (Qn 1)). != " + 3. M +. !+ 8i > 0 Qi + 1i . 5" " " + Si Qi 8i > 1 +: Si = fa 2 Qi j 9N =N(a): 8l > N 8k > 1 alk = 0 8l 6 N 8k > 1 alk 2 F 1i 1g: X= " " DEMi Pi Qi 8i > 1 : DEMi = fa 2 Qi j 9M = M(a): 8l > M 8k > 1 alk = al+1k+1 2 F 1i 1 8l 6 M 8k > 1 alk 2 F 1i 1g Pi = fa 2 Qi j 8k l akl 2 F 1i 1 g: 5 Pi = CFMX (F), Si = Soc(Pi ) = Soc(DEMi ). !+ $ 8m n, 2 6 m < n, + hmn +%+ hn 1n : : :hm+1m+2 hmm+1 : Qm ! Qn . 5 hmn (Sm ) " + Qn, "= 8m m , 2 6 m < m < n, hmn (Sm ) \ hm n (Sm ) = 0, hmn (Sm ) hm n(Sm ) hmn (Sm ). "

+ , Sm = Soc(Pm ), hmm+1 (Sm ) \ Sm+1 = 0 hmm+1 (Pm ) Sm+1 " " hmm+1 + " Sm . L + , 8m n, 2 6 m < n, hmn (Sm ) \ DEMn = 0 hmn (Sm ) DEMn hmn (Sm ). 5" " " Rn Qn: Rn = h1n(S1 ) + h2n(S2 ) + : : : + hn 1n(Sn 1) + DEMn : 5 S1 | Q1 = CFMX (F ), , h1n(S1 ) | Qn A " Soc(Rn ) = h1n(S1 ) Rn= Soc(Rn) = h2n(S2 ) + : : : + hn 1n(Sn 1 ) + DEMn = Rn 1 h2n(S2 ) + : : : + hn 1n(Sn 1 ) + DEMn

" h2n(P2 ) = Qn 1 . !, Soc(DEM1 ) = S1 , DEM1 = Soc(DEM1) = F(x) | " , . 5 +, "

", Rn | " n + 1 " 0 Soc1(Rn) = h1n(S1 ) Soc2 (Rn) = h1n(S1 ) + h2n(S2 ) : : : Socn (Rn) = h1n(S1 ) + : : : + hn 1n(Sn 1) + Sn Socn+1(Rn) = Rn, "= Rn i, " " . 8i, i 6 n ; 1, Rn= Soci (Rn) = != + , Rn | " V- . !+ + R0 " F(x). 5 R0 | " V- . E "= " + "

" n. ;

;

;

;

;

;

;

0

0

0

0

0

;

;

;

;

;

;

;

;

;

;

0

;

SV-

129

1 Rn | " V- . # $, Rn+1 | $ " V- . 1 M | "+ " " Rn+1- . X M Soc(Rn+1) = 0, M | " " Rn- , " ""$

M " Rn- . 5 " C /6, 1.13] /6, 6.17] M $

" Rn+1 - . X M Soc(Rn+1 ) 6= 0, M Soc(Rn+1 ) = M , +, M % + Soc(Rn+1)Rn+1 . 5 Soc(Rn+1 )Rn+1 | "" , M +% " Soc(Rn+1)Rn+1 . 5 +, $ , M | " Rn+1 . Rn+1 Qn+1 = CFMX (F ), Qn+1 | " . 5 M Qn+1 = M Soc(Rn+1 ) Qn+1 = M Soc(Rn+1 ) = M (

Soc(Rn+1 ) = Soc(Qn+1 )), M | " Qn+1. 5 " /6, 9.2] M | " Qn+1- " /6, 6.17] M | " Rn+1- . [ + .

1] G. Baccella. Generalized V-rings and von Neumann regular rings // Rend. Sem. Mat. Univ. Padova. | Vol. 72. | 1984. | P. 117{133. 2] G. Baccella. Semiartinian V-rings and semiartinian von Neumann regular rings. 3] G. Baccella. Von Neumann regularity of V-rings with Artinian primitive factor rings // Proc. Amer. Math. Soc. | 1988. | Vol. 103, no. 3. | P. 747{749. 4] N. V. Dung, P. F. Smith. On semiartinian V-modules // J. Pure Appl. Algebra. | 1992. | Vol. 82, no. 1. | P. 27{37. 5] L. Fuchs. Torsion preradicals and ascending Loewy series of modules // J. Reine Angew. Math. | 1969. | Vol. 239/240. | P. 169{179. 6] K. R. Goodearl. Von Neumann Regular Rings. | London: Pitman, 1979. | Monographs and Textbooks in Mathematics. 7] B. L. Osofsky. Rings all of whose nitely generated modules are injective // Pacic J. Math. | 1964. | Vol. 14. | P. 645{650. 8] C. Nastasescu, N. Popescu. Anneaux semi-artiniens // Bull. Soc. Math. France. | 1968. | Vol. 96. | P. 357{368.

' ( ) 1997 .

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Abstract S. Terzic, Cohomology with real coecients of generalized symmetric spaces, Fundamentalnaya i prikladnaya matematika, vol. 7 (2001), no. 1, pp. 131{157.

In the article we consider generalized symmetric spaces of the compact simple Lie groups. We give a classi2cation of these spaces and an explicit description of their algebras of cohomology with real coe3cients. In the case of such spaces of second category, the direct computation of their cohomology algebras is given.

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179

3# "1 (v p< x< t y) # ( G~ v (p)(x< t y) # Gv (p)] T(xty)(@Dk Dl ), # "1 = minf"1 (v p< x< t y) j 0 6 v 6 1 p 2 P (x< t y) 2 @Dk Dl g "2 = minfkrV Gv (p)(x< t y)k j 0 6 v 6 1 p 2 P (x< t y) 2 @Dk Dl V 2 T(xty) (@Dk Dl ) kV k = 1g " = (1=10) minf"1 "2 1g. rV - # ( V . C , # "^1 (v p< s x< y) (- HB v (p)(s x< y) Hv (p)] T(sxy) (Dk @Dl ) "^1 = minf"^1 (v p< s x< y) j 0 6 v 6 1 p 2 P (s x< y) 2 Dk @Dl g "^2 = minfkrV Hv (p)(s x< y)k j 0 6 v 6 1 p 2 P (s x< y) 2 Dk @Dl VB 2 T(sxy) (Dk @Dl ) kVB k = 1g "^ = (1=10) minf"^1 "^2 1g. 1( v v0 , $ # v ; v0 ( ( v ; v0 ), p 2 P Jvv0 (p)(x) x 2 @Dk Rm ( Kvv0 (p)(x< t y) Lvv0 (p)(s x< y) (x< t y) 2 @Dk Dl (s x< y) 2 Dk @Dl Rm+n - . 3# Mvv0 (p)(x) # ( # Rm, ( ( g~v0 (p)(x) g~v (p)(x), , vv0 (p)(x) | ( , ( Mvv0 (p)(x) ( (~gv0 (p)(x) g~v (p)(x)) # v ; v0 # , 0 6 vv0 (p)(x) < ). 3 0 M~ vv0 (p)(x< t y) MB vv0 (p)(s x< y) Rm+n, ( (~gv0 (p)(x) 0) (~gv(p)(x) 0) HB v0 (p)(s x< y) HB v (p)(s x< y) ( -), vv0 (p)(x< t y) (= vv0 (p)(x)) vv0 (p)(s x< y) | - ( , 0 6 vv0 (p)(s x< y) < ). A Jvv0 (p): @Dk ! SO(m R) Kvv0 (p): @Dk Dl ! SO(m + n R) Lvv0 (p): Dk @Dl ! SO(m + n R) (, - x, (x< t y) (s x< y) (- ( $ m ( ( m + n ( # , - Mvv0 (p)(x), M~ vv0 (p)(x< t y) MB vv0 (p)(s x< y) vv0 (p)(x), vv0 (p)(x< t y) vv0 (p)(s x< y) , | g~v0 (p)(x) g~v (p)(x),

180

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| (~gv0 (p)(x) 0) (~gv (p)(x) 0), | # HB v0 (p)(s x< y) HB v (p)(s x< y). O Mvv0 (p)(x) , Jvv0 (p)(x) # < , M~ vv0 (p)(x< t y) MB vv0 (p)(s x< y). > , ( Jvv0 (p): @Dk ! GL(m R) Kvv0 (p): @Dk Dl ! GL(m + n R) Lvv0 (p): Dk @Dl ! GL(m + n R) ' k J (p)(x) Jvv0 (p)(x) = kkg~g~v (p)(x) x 2 @Dk < v0 (p)(x)k vv0 k Kvv (p)(x< t y) Kvv0 (p)(x< t y) = kkg~g~v (p)(x) (x< t y) 2 @Dk Dl < 0 v0 (p)(x)k B v (p)(s x< y)k k l Lvv0 (p)(s x< y) = kkHH B v0 (p)(s x< y)k Lvv0 (p)(s x< y) (s x< y) 2 D @D : = , ( Jvv0 (p) Kvv0 (p) Lvv0 (p) ((-( C 1- # x (x< t y) (s x< y) . H , # v ; v0 , Pvv0 (p)(x< t y) p 2 P (x< t y) 2 @Dk Dl Rm+n. 3# M^ vv0 (p)(x< t y) # ( # Rm+n, ( ( Kvv0 (p)(x< t y)G~ v0 (p)(x< t y) G~ v (p)(x< t y), , vv0 (p)(x< t y) | ( , 0 6 vv0 (p)(x< t y) < ). = Pvv0 (p): @Dk Dl ! SO(m + n R) , (x< t y) @Dk Dl (- Rm+n, # M^ vv0 (p)(x< t y) vv0 (p)(x< t y), ( Kvv0 (p)(x< t y)G~ v0 (p)(x< t y) G~ v (p)(x< t y) ( M^ vv0 (p)(x< t y) , Pvv0 (p)(x< t y) # ). = Pvv0 (p): @Dk Dl ! GL(m + n R) ' kG~v (p)(x< t y)k Pvv0 (p)(x< t y) = kKvv0 (p)(x< t y)G~ v0 (p)(x< t y)k Pvv0 (p)(x< t y) (x< t y) 2 @Dk Dl . = , 0 ((( C 1 - # (x< t y).

181

E #, Pvv0 (p)(x< t y)Kvv0 (p)(x< t y)G~ v0 (p)(x< t y) = G~ v (p)(x< t y) Lvv0 (p)(s x< y)HB v0 (p)(s x< y) = HB v (p)(s x< y)

(1) (10 )

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k > 1, n > l > 1, Gkm++l!nkk+l | $ Gm+n k+l , ( ! (k + l)- !", ! !

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= Q ;! S m+n;1 Rm+n ;! Rm $ Q 0 2 Rm. ) S C 1 -$!, = C 1 -$! . !0 . 1 # 10 . 3# Vmk++ln!kk+1l+1 | ( T'( Vm+n k+l+1 #$ $ (k +l +1)- Rm+n, ( $ (R1 : : : Rk+l Rk+l+1 ) $ p(R1 ) : : : p(Rk+l), p: Rm+n ! Rm | (, k $ p(Rk+l+1 ) = 0. = !0 p~: Vmk++ln!kk+1l+1 ! Gkm++l!nkk+l -, (- (R1 : : : Rk+l Rk+l+1) # hR1 : : : Rk+l i, (- R1 : : : Rk+l. . !0 k Vmk++ln!kk+1l+1 , p~, ((( ( ) Gkm++l! n k+l !0 Vk+l k+l S n;l;1 . 1( ( S (Vmk++ln!kk+1l+1 )

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# k Q ;;;;! Gkm++l! n k+l . S ( , S (Vmk++ln!kk+1l!+10) # , 0 !0 s: Q ! S (Vmk++ln!kk+1l+1 ): A !0 p^: Vmk++ln!kk+1l+1 ! Sm+n;1 -, (-- (R1 : : : Rk+l Rk+l+1) Rk+l+1 . = = : Q ! S m+n;1 = = p^ S! s. . =(q), q 2 Q, # S(q), p(=(q)) = 0. 2 = k S: I P Dk @Dl ! Gkm++l! n k+l (( (v p< s x< y) # S(v p< s x< y), (- HB v (p)(s x< y) #- # Hv (p)] (T(sxy) (Dk @Dl )). = S ( , (H0(p) HB 0(p)) ( F^ (p) Dk Dl Rm+n. 3 0 # 10. = : I P Dk @Dl ! Sm+n;1 C 1 - (s x< y) 2 Dk @Dl , =(v p< s x< y) S(v p< s x< y) = I P Dk @Dl ;! S m+n;1 Rm+n ;! Rm I P Dk @Dl 0 2 Rm. > 0 , ( v v0 , $ jv ; v0 j 6 , $ p 2 P (x< t y) 2 @Dk Dl V 2 T(xty) (@Dk Dl ), kV k = 1, - (. G g~v0 (p)(x) g~v (p)(x) #4 (0 # ( Jvv0 (p)(x) (3) Kvv0 (p)(x< t y)). G Kvv0 (p)(x< t y)G~ v0 (p)(x< t y) G~ v (p)(x< t y) #4 ( # ( - (4) ( Pvv0 (p)(x< t y)).

184

. .

kKvv (p)(x< t y)G~v (p)(x< t y) ; G~v (p)(x< t y)k < 20" : kG~ v(p)(x< t y) ; Kvv (p)(x< t y)G~v (p)(x< t y)k < " min 1 kKuu (p0 )(x0< t0 y0 )G~ u (p0 )(x0< t0 y0 )k < 20 kKuu (p0)(x0< t0 y0)k 0 6 u u0 6 1 p0 2 P (x0 < t0 y0 ) 2 @Dk Dl : k(~gv (p)(x) 0) ; (~gv (p)(x) 0)k < " minfk(~g (p0 )(x0) 0)k j 0 6 u 6 1 p0 2 P x0 2 @Dk g: < 20 u krV Gv (p)(x< t y) ; rV Gv (p)(x< t y)k < 10" : kGv(p)(x< t y) ; Gv (p)(x< t y)k < < ("=100)(1=(maxfkrV 0 , ( v v0, $ jv ; v0 j 6 ^, $ p 2 P , (s x< y) 2 Dk @Dl VB 2 T(sxy) (Dk @Dl ), kVB k = 1, (# - (. G HB v0 (p)(s x< y) HB v (p)(s x< y) #4 ( # ( ( (30 ) Lvv0 (p)(s x< y)). kHB v(p)(s x< y) ; HB v0 (p)(s x< y)k < 10"^ minf1 kHB u(p0)(s0 x0< y0)k j 0 6 u 6 1 p0 2 P (s0 x0< y0 ) 2 Dk @Dl g: (40 ) krV Hv(p)(s x< y) ; rV Hv0 (p)(s x< y)k < 10"^ : (50 ) kHv(p)(s x< y) ; Hv0 (p)(s x< y)k < < (^"=100)(1=(maxfkrV =(v0 p0< s0 x0< y0 )k j 0 6 v0 6 1 p0 2 P (s0 x0< y0) 2 Dk @Dl VB 0 2 T(s x y ) (Dk @Dl ) kVB 0 k = 1g)) (60 ) ( , (). . ^ , Hv (p) F^ (p). , (5), (6) (7) , ( v 6 kG~ v(p)(x< t y) ; G~ 0(p)(x< t y)k < 10" (10) 0

0

0

0

0

0

0

kKv0(p)(x< t y) ; IdRm n k k(~g0(p)(x) 0)k = " k(~g (p)(x) 0)k = k(~gv (p)(x) 0) ; (~g0(p)(x) 0)k < 20 0 " kKv0(p)(x< t y) ; IdRm n k < 20 kPv0(p)(x< t y) ; IdRm n k kKv0(p)(x< t y)G~ 0(p)(x< t y)k = = kG~ v (p)(x< t y) ; Kv0(p)(x< t y)G~ 0(p)(x< t y)k < " kKv0(p)(x< t y)G~ 0(p)(x< t y)k < 20 kKv0(p)(x< t y)k 1 kPv0(p)(x< t y) ; IdRm n k < 20" kKv0(p)(x< t y)k kPv0(p)(x< t y)Kv0(p)(x< t y) ; Kv0(p)(x< t y)k < 20" :

185

+

+

+

+

3 0 (11) kPv0(p)(x< t y)Kv0(p)(x< t y) ; Id m+n k < 10" : , (40 ) , ( v 6 ^ kLv0(p)(s x< y) ; Id m+n k kHB 0(p)(s x< y)k = "^ kHB (p)(s x< y)k = kHB v (p)(s x< y) ; HB 0(p)(s x< y)k < 10 0 kLv0(p)(s x< y) ; Id m+n k < 10"^ : (70 ) - ( ( 2]. 2. ( s0, s0 < 1, ! % $ p 2 P s 2 s0 1], (x< t y) 2 @Dk Dl F (p)(s x< t y) ; F(p)(1 x< t y) 6 4 krsF(p)(1 x< t y)k: s0 ; 1 3 1( ( # # 0 . . 1 # 2. , (

F (p)(s x< t y) ; F(p)(1 x< t y) rsF(p)(1 x< t y) = slim !1 s;1 0 P , # s0 , s0 < 1, ( s 2 s0 1] F (p)(s x< t y) ; F(p)(1 x< t y) ; rsF (p)(1 x< t y) 6 1 krsF (p)(1 x< t y)k: s;1 3 3 # F (p)(s x< t y) ; F(p)(1 x< t y) 6 4 krsF(p)(1 x< t y)k: s;1 3

R

R R

186

. .

1( s0 6 s 6 1 F(p)(s x< t y) ; F(p)(1 x< t y) F(p)(s x< t y) ; F (p)(1 x< t y) 6 : s0 ; 1 s;1 G $ $ . 2 # 4 ( - 0 . 2 . ( t0, t0 < 1, ! % $ p 2 P t 2 t0 1], (s x< y) 2 Dk @Dl ^ ^ x< t y) ; F(p)(s x< 1 y) 6 4 kr F(p)(s F(p)(s x< 1 y)k: 3 t ^ t0 ; 1 = (# 2, s0 , 1=2 < s0 < 1, , ( $ v 6 , p 2 P, s 2 s0 1], (x< t y) 2 @Dk Dl , V 2 T(xty) (@Dk Dl ), kV k = 1, (# - (: F(p)(s x< t y) ; F (p)(1 x< t y) < 2krsF(p)(1 x< t y)k< (12) s0 ; 1 krsF (p)(s x< t y) ; rsF (p)(1 x< t y)k < 10" < (13) (14) krV F (p)(s x< t y) ; rV F(p)(1 x< t y)k < 10" < kF(p)(s x< t y) ; F (p)(1 x< t y)k < < ("=10)(1=(maxfkrV (Pv 0(p0 )(x0 < t0 y0 )Kv 0(p0 )(x0< t0 y0 ))k j 0 6 v0 6 1 p0 2 P (x0< t0 y0 ) 2 @Dk Dl V 0 2 T(x t y ) (@Dk Dl ) kV 0k = 1g)) (15) ( (15) , 0 (). = , # s0 . 3# s1 = s0 + (1=3)(1 ; s0 ). C , #( 20, # t0, 1=2 < t0 < 1, , ( $ v 6 ^, p 2 P , t 2 t0 1], (s x< y) 2 Dk @Dl , VB 2 T(sxy)(Dk @Dl ), kVB k =1, ( F(p)(s ^ ^ x< t y) ; F(p)(s x< 1 y) < 2kr F(p)(s x< 1 y)k< (80 ) t^ t0 ; 1 "^ < ^ ^ krtF(p)(s x< t y) ; rtF(p)(s x< 1 y)k < 10 (90 ) "^ < ^ ^ krV F(p)(s x< t y) ; rV F(p)(s x< 1 y)k < 10 (100) ^ ^ kF(p)(s x< t y) ; F(p)(s x< 1 y)k < < (^"=10)(1=(maxfkrV Lv 0 (p0)(s0 x0< y0 )k j 0 6 v0 6 1 p0 2 P (s0 x0< y0 ) 2 Dk @Dl VB 0 2 T(s x y ) (Dk @Dl ) kVB 0 k = 1g)) (110) ( , (). 0

0

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187

= , t0 . 3# t1 = t0 + (1=3)(1 ; t0 ). 1 # C 1 - ' (s), (t), (s) (t) 0 1] , (# - (: (s) = 0 0 6 s 6 s1 < (16) (t) = 0 0 6 t 6 t1< (120) (1) = 1 0 (1) = 0< (17) (1) = 1 0 (1) = 0< (130) j(s)j 6 1 j 0(s)j < 1 ;2 s0 < (18) j(t)j 6 1 j0(t)j < 1 ;2 t0 < (140) (s) = 0 0 6 s 6 s0 < (19) (t) = 0 0 6 t 6 t0< (150) (1) = 0 (1) = 0< (20) (1) = 0(1) = 0< (160) j0(s)j > 10j 0(s)j s1 6 s 6 1< (21) 0 0 j (t)j > 10j (t)j t1 6 t 6 1< (170) j(s)j 6 20< (22) j(t)j 6 20: (180) ( M(v) = maxfkGv (p)(x< t y) ; G0(p)(x< t y)k j p 2 P (x< t y) 2 @Dk Dl g N(v) = maxfkHv (p)(s x< y) ; H0(p)(s x< y) j p 2 P (s x< y) 2 Dk @Dl g: , - Fv (p) = (Fv (p) fv (p)) F^v (p) = = (F^v (p) f^v (p)), p 2 P , ( v 6 v 6 ^ - : Fv (p)(s x< t y) = Id m+n +(s)(Pv0 (p)(x< t y)Kv0(p)(x< t y) ; Id m+n )] (F (p)(s x< t y) ; F(p)(1 x< t y)) + (s)(Gv (p)(x< t y) ; G0(p)(x< t y)) + + (s)M(v) 10X0 0 0 ! U A A0. '$ ! A + B + + XU + X U A + B " . .#, #( , #, Fv (p) ((( . H (33), #, rsFv (p)(s x< t y) rW Fv(p)(s x< t y) . 3 , , 5, # 3 4 A = G~ v (p)(x< t y) A0 = rW Gv (p)(x< t y): 3 0 ( 5 (-(, #(, #( ". 3 4 ( #, < 1, #( F (p) F (p). 5 -- - ( v 6 2. 3 ( , -- - ( $ v 2 0 1]. 5 - # (i). 3 # . 1 , F^v (p)(s x< t y) ((( C 1 - (s x< t y)< (240) ^ F^0(p) = F(p)< (250) F^v (p)(s x< 1 y) = Hv (p)(s x< y)< (260) (270) rtF^v (p)(s x< 1 y) = HB v (p)(s x< y)< F^v (p) ((( . (280)

192

. .

, , (, # ' (190) ((-( C - , (240). (250) (( . , (190), #( (130) (160), , H0(p)(s x< y) = ^ = F(p)(s x< 1 y), # (260). ^ (270) (220 ), (130), (10), (160) rtF(p)(s x< 1 y) = B = H0(p)(s x< y). (280) # , rV F^v (p)(s x< t y) 6= 0 ( $ BV 2 T(sxty)(Dk Dl ). / 4 VB VB = VBt + VB(sxy) , VB(sxy) | ( VB T(sxy)(Dk @Dl ), VBt | ( VB (-, #- T(sxy) (Dk @Dl ) T(sxty) (Dk Dl ). . rV F^v(p)(s x< t y) = Bt rtF^v(p)(s x< t y) + B(sxy)rW F^v (p)(s x< t y) (290) WB = VB(sxy)=kVB(sxy) k, Bt B(sxy) | $ ( (,

$ -. 3 . ( ! BB0 Rm+n, kBB0k < "^, !" % rW F^v (p)(s x< t y) = rW Hv(p)(s x< y) + BB0 : . 1 # 30. , (140) (110) , "^ : ^ k(t)rW Lv0(p)(s x< y)(F^(p)(s x< t y) ; F(p)(s x< 1 y))k < 10 = kId m+n +(t)(Lv0(p)(s x< y) ; Id m+n)] 2^" ^ ^ (rW F(p)(s x< t y) ; rW F(p)(s x< 1 y))k < 10 (140 ), (70 ) (100). , #( (140) (50 ), k(t)(rW Hv(p)(s x< y) ; rW H0(p)(s x< y))k < 10"^ : , (180) (60) k(t)N(v)rW =(v p< s x< y)k < 102^" , , (50 ) F^ (p)(s x< 1 y) = H0(p)(s x< y) , krW F^ (p)(s x< 1 y) ; rW Hv(p)(s x< y)k < 10"^ : 3 0 (230 ) ( 30 . 2 4 . ( ! BB , UB , UB 0 Rm+n, ! BX, XB 0, ! % kBB k < "^, XB > 10XB 0, kUB k = kUB 0k = 1, ! UB ! S(v p< s x< y)

rtF^v(p)(s x< t y) = HB v (p)(s x< y) + BB + XB UB + XB 0UB 0: 1

0

R

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193

. 1 # 40. , (220) , rtF^v(p)(s x< t y) = 0(t)Lv0(p)(s x< y) ; Id m+n] ^ ^ ^ (F(p)(s x< t y) ; F(p)(s x< 1 y)) + rtF(p)(s x< 1 y) + + (t)Lv0(p)(s x< y) ; Id m+n ]rtF^ (p)(s x< 1 y) + + Id m+n +(t)(Lv0 (p)(s x< y) ; Id m+n )] ^ ^ (rtF(p)(s x< t y) ; rtF(p)(s x< 1 y)) + 0 0 0 ~ B B + (t)N(v p< s x< y)U + (t)N(v)U

R

R

R

R

~ p< s x< y) = kHv (p)(s x< y) ; H0(p)(s x< y)k N(v ~ p< s x< y) UB 0 = (Hv (p)(s x< y) ; H0(p)(s x< y))=N(v UB = =(v p< s x< y): ~ p< s x< y) 6 N(v). = , 0 6 N(v ^ , #( (140 ), (80 ), (10 ), (40) rt F(p)(s x< 1 y) = = HB 0(p)(s x< y), , 4^" ^ k0(t)Lv0(p)(s x< y) ; Id m+n ](F(p)(s x< t y) ; F^ (p)(s x< 1 y))k < 10 (40) "^ : ^ krtF(p)(s x< 1 y) ; HB v (p)(s x< y)k < 10 , (140), (10 ), (40 ) , "^ ^ k(t)Lv0(p)(s x< y) ; Id m+n]rtF(p)(s x< 1 y)k < 10 (140), (70) (90 ) kId m+n +(t)(Lv0(p)(s x< y) ; Id m+n)] 2^" : ^ ^ (rtF(p)(s x< t y) ; rtF(p)(s x< 1 y))k < 10 G (170) $ 4 . 2 1 , F^v (p) ((( . 1( 0 #, rtF^v (p)(s x< t y) rW F^v (p)(s x< t y) (. (290)), 5, # 30 40 - ( A = HB v (p)(s x< y) A0 = rW Hv (p)(s x< y) B U 0 = UB 0 X = X B X0 = XB 0: B B 0 = BB 0 U = U B = B , #( "^, #(, ( 5 (-(.

R

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194

. .

> , #( (210), (240 ){(280 ), , ^ : E ! C ((( . (210), (240), (280) - # ( F^v : P ! E , 0 6 v 6 ^, ( F^ ( (250)), (260) (270) -, F^v Hv . O ^ < 1, (, #( F^^(p) F^(p), -- - ( v 6 2^. 3 ( , -- - F^v (p) ( $ v 2 0 1]. , , (i) #- . 3 (ii) - (. (ii)0. ~ : C ! D (m > k, n > l)

.

1 # 0 ( #- ( # # (i), ( ( ( : E ! B. O , 0 C D # # (, ($, $ 0 B E . 3 0 , ( # , 4# 0 . 3# ( Gv : P ! D, 0 6 v 6 1, P | #

0, Gv (p) - 4 Gv (p) = ((Gv (p) gv(p)) (G~ v(p) g~v (p)) GB v(p)): 1 , # v = 0 Gv B H : P ! C H(p) = ((H(p) f(p)) H(p)) p 2 P . . G0 = ~ H. G , ~ ((( , , ( Hv : P ! C 0 6 v 6 1 Hv (p) = ((Hv(p) fv (p)) HB v(p)) p 2 P - ( Gv , . . ~ Hv = Gv , v = 0 H0 = H: 5 Hv (p) fv (p) Hv # ( Fv (p) fv (p) # (i). 3 - HB v (p). 7 -- $ $ , (- 0 Gv (p) Gv ( p P : (qGv (p)) (T(Rn))

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195

q: Rm+n = Rm Rn ! Rn ( # (, ( # $ ((-( (, ( ( ( (Gv (p))! '' ( Gv (p). 7 # (

T(@Dk @Dl ) ! @Dk @Dl (qGv (p)) (T(Rn)) ! @Dk @Dl @Dk @Dl ( l ; 1), 0 # 0$ , ((-( (qGv (p)) (T(Rn)) ! @Dk @Dl : , Pv (p)(@Dk @Dl ) ! @Dk @Dl # -. H , 0 . > , # # . C # Pv (p)(Dk @Dl ) ! Dk @Dl # (qHv (p)) (T(Rn)) ! Dk @Dl -

T(Dk @Dl ) ! Dk @Dl (qHv (p)) (T(Rn)) ! Dk @Dl : . Hv (p) Gv (p) ~ , ( Pv (p)(Dk @Dl ) ! Dk @Dl @Dk @Dl Pv (p)(@Dk @Dl ) ! @Dk @Dl : 1 , ( Gv (p) Hv (p) 0 6 v 6 1, v, # ( P(p)(I @Dk @Dl ) ! I @Dk @Dl P(p)(I Dk @Dl ) ! I Dk @Dl ( $ fvg @Dk @Dl fvg Dk @Dl - Pv (p)(@Dk @Dl ) ! @Dk @Dl Pv (p)(Dk @Dl ) ! Dk @Dl .

196

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> , P(p)(Dk @Dl 0@Dk@Dl I @Dk @Dl ) ! Dk @Dl 0@Dk@Dl I @Dk @Dl (( ( P0 (p)(Dk @Dl ) ! Dk @Dl P(p)(I @Dk @Dl ) ! I @Dk @Dl P0 (p)(@Dk @Dl ) ! @Dk @Dl : = , ((( ( P(p)(I Dk @Dl ) ! I Dk @Dl Dk @Dl f0g@Dk@Dl I @Dk @Dl . 3 -

v (p) m+n @Dk @Dl G;! R n f0g Rm+n ;! Rm @Dk @Dl 0 2 Rm, 0 #, GB v (p)(Z), Z 2 @Dk @Dl , ( (qGv (p)) (T(Rn)) ! @Dk @Dl Z. 1 , GB v (p)(Z) # Z Gv (p) (=2 (Gv (p)) TZ (@Dk @Dl )). 3 0 , -- Pv (p)(@Dk @Dl ) ! @Dk @Dl GB v (p) # 0 (. = 2B (( GB v (p), 0 6 v 6 1, v, G(p) ( P(p)(I @Dk @Dl ) ! I @Dk @Dl : C (, - - HB v (p), 0 6 v 6 1, # ( Pv (p)(Dk @Dl ) ! Dk @Dl : B ( = 2 ( HB v (p) v, H(p) P(p)(I Dk @Dl ) ! I Dk @Dl : B GB 0 (p), HB 0(p) G 0 (p) G(p) B ( HB 0(p) G(p) P(p)(Dk @Dl 0@Dk@Dl I @Dk @Dl ) ! Dk @Dl 0@Dk@Dl I @Dk @Dl : . I Dk @Dl ' ( Dk @Dl f0g@Dk@Dl I @Dk @Dl , B # ( H(p), B HB 0(p) G 0 (p) G(p) ( - - HB v (p). > 0 4 ( # . 2

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197

1] S. Smale. The classication of immersions of spheres in Euclidean spaces // Ann. of Math. | 1959. | Vol. 69. | P. 327{344. 2] S. Smale. Regular curves on Riemannian manifolds // Trans. Amer. Math. Soc. | 1958. | Vol. 87. | P. 492{512. 3] S. Smale. A classication of immersions of two-sphere // Trans. Amer. Math. Soc. | 1958. | Vol. 90. | P. 281{290. ' ( ) 1997 .

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Abstract R. Hildebrand, Classication of phase portraits of optimal syntheses, Fundamentalnaya i prikladnaya matematika, vol. 7 (2001), no. 1, pp. 199{233.

The paper is devoted to the investigation of controllable oscillating systems of ordinary di1erential equations a2ne in scalar nonsymmetric control in a neighborhood of a singular point of focus or center type. Integrands in value functionals are quadratic in phase coordinates. We classify such systems in case of general position by arising optimal syntheses. The existence of optimal synthesis is proved and its structure is described.

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, 2 0 / 2 0 1 2 , ! . 7]. 5 :; , 2 0 u 0 0 / U . 6 2 2 0 , 0 0 U . 4 && ! && / !( . 4 & Z1 J = (F (x) + uG(x)) dt ! min (1) 0

x(t) = x~: (2) x_ = A(x) + B (x)u@ x 2 U R2 u 2 0 1] t!lim +1 5 2 0 & x~ 2 R2. U x~ . ,

u(t) | & t, A B &

F G 0 C 3. A

x~ 2 & : 2 @A @x (~x)

-01

201

2 A(~x) = 0. B U 0 . * 0

5 &

F x~ & , . . & F 1 x~ !. > G 0 x~. B

x(t) / U . B !

!( / ! & , Z1 x_ = y I = F (x) dt ! inf 0

y_ = ;x + u u 2 0 1]: 5 2 (0 0), 1 0 2 . . , 2 0 x y 0, u 0, 0 / . 1. C & 2 F (x) 1. D ,

, 201 . 1 . : a 0 1 (;1 0). 0g fx j dAB (x) < 0g . f(x ) j x = 0 1 = 0 2 < 0g f(x ) j x = 0 1 = 0 2 > 0g. 7 "

| " (x = 0 = 0). . 6 T X H0 H1. ) (7) , dAB ;&& !. 6 ;

B2 F ; A2 G + B2 H0 ; A2 H1 0+F = 1 = d 1 ;BB21 ;AA12 H H1 + G dAB ;B1 F + A1 G ; B1 H0 + A1 H1 AB (9)

fH0 = 0 H1 = 0g, !( M, ;

= d 1 ;BB2 FF;+AA2 GG : (10) , $ +

AB

1

1

) (10) 1 M+ M; . /

T X 2! fx = 0 1 = 0 2 6 0g fx = 0 1 = 0 2 > 0g . 6 dAB = 0 2 ;&& (7) , . . / : A1(G + H1) ; B1 (F + H0) = A1 (G + H1) ; B1 (F ; uH1) = 0: (11) A G H1 = 0, F = B11 (7). 6 x~ . 40 (7) x~ fx = 0 1 = 0g. ! 2 . 0 , / / AB . 6 3 ! AB 0 x2, / &

F AB x~

dxdF2 = 0, d2 F2 = @ 2F2 = ; cos . ! F = ; 1 cos x2 + (r3 ). 2 2 dx2 @x2 3 A G 3 1 , B1 = (r ). 5 2 ( 2 2 ),

cos < 0. , F ; AB11G = X(x22 ) 6= 0. ) 0 2 . 2 6 2

, 0( M. ( ( . 10]) 2/ 0 . &&

0 @H @u 0 1 . G &

H = H0 + uH1,

ux ) = 0 adh H , ad0 H = H , adn+1 H = fH adn H g. ( dtd )h @H (@u 1 1 H 1 H 1 H 1 H 8 f g 2 2 . B H_ 1

1 (7)

218

.

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t

= 0, H 1 2 = u @@xF21 = u sin . H

sin 6= 0, ; 2 2 0 x~ 1. . u H1 1 = 0. ! u 0, , x x~, 0. 2 2 M+ , M; . 6 0 2

dAB 6= 0. (9) (12). &&

; A ; uB 2 2 _ dAB H1 = A1 + uB1 A B ] H1 +

B F ; A G 2 2 + ;B F + A G A B ] ; dAB DA G + dAB DB F 1 1 H1 . 2 C , A + uB | Q. D ,

;Q2 A B ] = d_ ; d tr @Q : AB AB @x Q1 6 @Q _ _ dAB H1 = dAB ; dAB tr @x H1 + C: (13) D M+ M;

H1 = 0. , ; / 2 / C , 2 0 C = 0. H_ 1 = dAB 3. C C2 ; C = 21 sin x21 + cos x22 + (x) (14) @ 2 2 = (r) = (r3 ) r = (r2 ) @x 2 ( 32 ) (14) C < 0 U n fx~g 2 ( 2 ) (14) C x~ C1 x~ x~ x~ C2 X H_ 1 = fH H1g = fH0 H1g =

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! _AB _ d @Q d _ 1 + C_ ; C d2_AB = 1 H1 = dt d ; tr @x H1 + ddAB H_ 1 ; tr @Q H @x d d AB

AB

AB

AB

C_ : (15) + = ; dC tr @Q AB @x dAB D / ^Ci

C = 0, (15) (14) / H1 1 = Dd A C + u Dd B C = d 1 A1 sin x1 + A2 cos x2 + (r3 )+ u(sin x1 + (r2 ))]: AB AB AB :;&& u , 2 0

2 1. ) H1 1 = 0 u = ; DDBA CC = ; x1 ; x2 + + ctg x2 + (r2 ). H , 0 / 2 0 1]. 60 (14) !, ; ctg = ; xx221 + (r) u = ; x12 (x21 + x1x2 + x22) + (r2 ) < 1. 6 2 (;2 2)

, u > 0 x2 < 0. 5 2 , 2 0 ! ^C 3 ^C 4 . D

C 3 & 2 0 x~, C 4 | . 2 0 / 2 .

6. ' ) (11) 1 & ! H1 AB . C , ; AG e1 i = hBF ; AG V i : H1 = ;A1 GQ+ B1 F = hBFhQ (16) e1 i hA + uB V i 1 6 BF ; AG Q = A + uB e1 = (1 0) 0 !2 V , Q. 5 2 , (13) / AB ! 2 . C ; 2 u = 0. u = 0 2 (16) V = x? = (;x2 x1), H1

i H1(t) = ;G(t) + F (t) hhBx Ax i (t) + dAB (t)f (t). ; 0 (13), ?

?

220

.

0 / !( ! & ! f (t): rF i f_ + hr Aif + hx?F Ai2 hx? A ; Dx Ai + hhx x? Ai = 0:

2 s(t),

Zt ; Rt hrAi d ; x B + x B 2 1 1 2 H1(t) = ;G + F ;x A + x A + dAB ; e s( ) d ; 2 1 1 2 0

1 ;G + F ;x2B1 + x1B2 (0) : dAB (0) ;x2 A1 + x1A2 6 . H

dtd (;x2 A1 + x1 A2) = hr Ai(;x2A1 + x1A2 ) + Rt ; hrAi d

;e

0

tR2 hrAi d t

e1 = ((;;xx22 AA11 ++xx11 AA22 )()(tt21)) + (r). hx rF i = 2F + (r3 ). 5 2 , s = 2?F

: , A ; Dx A = (r2 ), hx Ai + (r). ,

+ (r3 )

Zt 1 H1(t) = ;x A + x A ;Fx2 + dAB ; 2F d ; ;dFx2 (0) + (r2 ): (17) 2 1 1 2 AB 0

C T X H0: X x_ = A _ = rF ; rA : (18)

(2), (8), u = 0. C ! ^ (t) (18), 0(! &

H0. 1

+ ; = X ^ AB AB t+ t; . 6 H1(t+ ), H1 (t; ). H0 = 0 (11), H1 = ;G + BA11F = ; cos2 x2 + (x22 ). . H1(t+ ) > 0 H1(t; ) < 0 2 6 1 U (dAB ), = ; xx12 . !, Ox2 . .& / . C 0 K = f(x1 x2) 2 U j ; xx12 2 ]g fx~g. G U , 0 K AB x~.

x~ 0 K 0 K+ = f(x1 x2) 2 K j dAB (x1 x2) > 0g, K; = f(x1 x2) 2 K j dAB (x1 x2) < 0g. D 0 / K+ K; (x1 x2) $ (dAB ) . x~

(dAB ) f0g ]. D , K x1 x2 &

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.

221

0 ! 2 0 K+ K; 2! 2 M+ M; T X . D ; / / (10) ! &

x , /, &

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(10). : &

/ dAB 0 C 3 0 K

! (r2 ) = (d2AB ). ) 0 Y . 2 C 0 KV 2 M, (

0 K+ K; M (x = 0 = 0). 0 ! 4 1 & (dAB ) C 2. C X x_ = A + B _ = r(F + G) ; ( r(A + B )) (19) 8"

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H0 + H1. u = 1 (2), (8). H

(19), /( 0 KV t = 0. 0g. 6 6 tV > t+ . 5. t 2 t; 0) H1(t) < 0 . 0 , ( t0 2 (t; 0), H1(t0 ) > 0. F 2( 0 , H1(t0 ) = 0 H1 (t) < 0 / t 2 (t0 0). D C (t0) < 0, dAB (t0) < 0, C > 0. . 6 ; H_ 1(t0 ) = dAB . 2 5 2 ,

tV < t; . H1( tV) = 0 (17)

F = 12 (sin x21 ; cos x22) + (r3 ) = X(r2 ), dxAB2 (0) = X(1) dAB = x1 + (r2 ), 0 = ;X(r;2 )f;x2( tV)X(r2 )+ x1 ( tV)X(r2 )g + (r2 ) =) x2 ( Vt ) = X(1)x1 ( Vt ): (24) 5 2 , ( tV) 0 . (t)

^ C 1 C 4 t1 , t4 . 6. t 2 t1 t4] H1(t) = ;X(r) . D ^C 4

x2 = ;X(r), x1 = X(r). (17), H1(t4 ) = ;&(1r2 ) fX(r3 ) + X(r)X(r2 ) + X(r2 )]g + (r2 ) = ;X(r). t 2 t1 t4). C (13). 2 ;&& { = H1 { . Q = A, t t4]

dAB 2 _ 1 +(r ) ; tr @A = x2 + (r) = (1). D ; (13)

= ;A2 BdAB @x x1

2 , 0 H1(t)

{ R t { d Zt4 ; R dAB d C H1(t) = e t4 dAB ; e t4 d + H ( t ) 1 4 : dAB +

+

.

.

t

D t t4]

C > 0, ; / 2/ 0

(r). 5 H1(t4 ) = ;X(r), 2 ,

H1(t), ! ;X(r). 2 7. C ( Vt ) = ;X(r2 ) .

224

.

. 6 (24) ( Vt ) 0 Vt < t4 . Rt1 5 0 6 , tV < t1 , H_ 1(t) dt = H1(t1) = ; X(r). t' D H_ 1 = (r), ; t1 ; Vt = X(1). , C (t1 ) = 0, , (24)

C_ = X(r2 ) tV t1]. ! Rt1 C ( tV) = ; C_ (t) dt = ; X(r2 ). 2 't

8. 4 " p & ^C 3 , V) " " 10 " C 2, ! " ( t & " "! " x ~

C 3 ) 0 8

2 / && / . &

0 7. 5 (24). dx2 9. 10 x ! x~ dx1 . D , dx2 10 , dx 1 dx2 = x2 + (r) = X(1): (25) dx1 x1 , ! 10 0 x1 . Hd xx12 x2 1 x2 2 (25), dx1 = ( dx dx1 ; x1 ) x1 = (1). 0 t 0, C_ < 0, dAB < 0. ; 2

H1 = H_ 1 = 0, H1 1 = dAB 2( 0 , H1(t0 ) = 0 H1(t) > 0 / "

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C < 0. t 2 (t0 0). D C ((t0)) > 0, dAB ((t0 )) < 0, ; H_ 1(t0 ) = dAB . 2 6 0 2 ! 10

^10 M+ . B , p | ^10 (x = 0 = 0), ^ (t) | (19), ^ (0) = p. 2 ! X ^ . H H1 & ! t

^ . 11. H1(t) > 0 t 0. , t (dAB ) > 0 AB dAB > 0. D , / (19), /( / 0 KV t = 0, & H1(t) / 0 t . D H_ 1 < 0 ^10, ; ( " > 0, H1(t) > 0 t 2 (;" 0). 2 U . ) 0 11 . 2 4 U & ! (x), !(! ! :; 10] 2 0 0 10 11 | . 12. (x) U C 3 ; 10 x~ C 3 ; 10 . C ! & , (! C 3 , 10 x~. 6 0 3 9 0 PC 1 / x~. 6 (dAB ) 1 & (dAB ). D , /

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226

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. C ^ M, (

^C 3 , ^10

(x = 0 = 0). : ^ 1 (10) 0 4 . C & : d (dAB ) = (dAB (dAB )). H

d dd AB = @d@ AB + @ lim!0 @ @ ddAB . 6 dAB @ = 0, d = (1)

lim d = lim @ . !(x) U ! F . ; . > / 2 0 x~ 2 , 1 x_ 2 = (x2 ). 5 . 8"

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227

B & / 2 & ~. ~ : (;2 2) ! ( 32 ) !( : q q p ~)2p+2 ctg ~ (1 ; ctg ~)2 + 2 ctg ~ + ctg ~ p1 2 +arcctg (1pctg 2 ctg ~ 4 4 q ;e = 0 q 2 2 ~ ~ ~ (1 ; ctg ) + ctg ; ctg

6= 0@ q ~ ctg ~ 2 ~ ; ; arcctg ctgq ; 1 = 0 = 0: ctg ; 1 2 ctg ~ ;

;

;

. arcctg (0 ). ) ! ~ ! ! & ! . 4 0 ,

2 1,

4 0 T X & ! F . 2 ( 23 ). 5 0 3

C = ;X(r2 ). | 0 . !, ! 6. 2 0 K; , (! , K0 . 0 . 6 0 2 0 K0

K^ 0 2 M; . K0 K^ 0 dAB0 = dAB , 0 = . H 0 , ; ! 0 / K0 K^ 0. C

(19), /( t = 0 0 K^ 0. 0 &

H0 + H1 . ;

t, / | dAB0 0 . ! &

t

H1(dAB0 0 t (dAB0 0 )) 0. ; 0 f(dAB0 0 t (dAB0 0 ))g 2 / K^ 1 2

M. H ; /. p0 | 0 K0 , !( dAB0 0 .

p^0 2 K^ 0 | p0 M; , ^ (t) | (19), /( t = 0 p^0. 6 & dAB p^1 = ^ (t ) dAB0 0 . 2 / dAB1 1. 1 (23)

Rt Rt dAB1 = dAB0 + d_AB dt = dAB0 + (1 + (r)) dt = ; dAB0 (ctg 02) + (dAB0 0), 0 0 ! dAB ! 0. B

,

2 C 1 , dAB 0 0

Zt x2(t ) = x2(0) + x_ 2 dt = x2 (0) + (r2 ):

0

(26)

228

.

,

(0) + (r2 ) tg + o(1): 1 = ; xx2((tt )) = ; ;d x2(ctg = ; (27) 2 0 ) + o(r) 0 1 AB0 6 , tg > 0, t = X(r), dAB1 = X(r), 1 = ;X(1). , K1 / K^ 1 &! 0 C > 0, ; H1(t) > 0 . D / K^ 0

H_ 1 = dAB 0 / t 2 (0 t ). ^ (t) | (18), /( p^1 2 K^ 1 t = 0. Vt = C < 0, ; tV > 0. = inf ft > 0 j H1(t) > 0g. 6 p^1

H_ 1 = dAB 1 2 ! X ^

^ &! . 5 2 x~ . ; ; t; > 0 1 ! AB , + t+ > t; | ! AB . 6 6

tV < t+ . 14. Vt > t; B 0 5. H 0 14 , p^2 = ^ ( tV) 0 2

M; . 2 1 ! ( tV) p2 . 4 tV 1 H1( tV) = 0. > H1(t) (17). 0 2 ! Fx2( tV) + (r4 ) = dAB ( tV)(r2 ). F = x1 (r) ; 21 cos x22 + (r3 ), dAB = x1 + (r2 ), x32( tV) + (r4 ) = x1( tV)(r2 ). ! ; xx21 (( 't't )) = (1). dAB2 = dAB (x1( tV) x2( tV)), 2 = ; xx21 (( tt'')) . 4 20 P : (dAB0 0) 7! (dAB2 2 ) K^ 0 M; M;. D , dAB2 = X(dAB0 ). , lim d = 0. dAB !0 AB2 .

0

. 2 & P & f0g ]. >

dAB1 dAB2 0 1 . > ! 1(dAB0 0 ) 0 0 1 (0 0) = ; tg0 (27). , & 2 (dAB1 1) 0 0 . ! 0 . Z, 20 P && dAB0 < 0. AB2 2 ) , @@ ((ddAB 0 0 ) 0 0! f0g ]. ; @dAB2 = dAB2 + (r) = X(1), @2 = C (p0 ) + (r) = X(1). @dAB0 dAB0 @0 C (p2 ) 5 2 , & 2 , f0g ], && !( & 0. 6 (27) lim 1(0 0 ) = ;1, !0 0

229

lim (0 0) = 0. H (17) (dAB1 ! 0) () (dAB2 ! 0). !, 0 !1 1 d2 > 0, lim (d ) = ;1, lim (d ) = +1. d 1 1 !;1 2 AB1 1 1 !+1 2 AB1 1 5 2 ,

lim (0 0) = 2 max lim 2 (0 0) = ;1: 0 !+1 2 0 !0 . 2 max | ( . . 5). 2 6 2 6

2 max

2 max 2

q q

q

-

0 1 0 0 2

0

-

0

$. 5

4 0 , ( 0 20 0 7! 2 (0 0). 6 ; (0 ) | 0 20 P. 2 ,

1 T X . G & &

2 (0 0) 0 & 2 = 0 ( . . 5 ), 2 = (0 0) . G

0 , 2 = 2(0 0 ) > 0 , ( ! 2 = (0 0) ( . . 5 ). 6 , / / 0 ( . C & ! A2(x). G1 x~ (3) ;e1 = (;1 0). ; &

A2 2 ! C 3, /(! x~ !(! Ox2 . s | 0 , 0( ; . C ! x_ = A, /(! s t = 0. 6 (3) 2 x~ . H ! x2 & ! t. H

x_ 2(0) = A2 (s) = 0, x2 (0) = x2min . t+ = minft > 0 j A2 ((t)) = 0g, t; = maxft < 0 j A2((t)) = 0g, x2max = min(x2(t+ ) x2(t; )). , x2max > 0 > x2min , 0 x^2 2 (x2min x2max) ( ! + 2 (0 t+ ), ; 2 (t; 0), x2( + ) = x2 ( ; ) = x^2. 6 1 2 x+1 = x1(( + )), x;1 = x1(( ; )).

230

.

R + C I (^x2 ) = F ((t)) dt

& ! x^2. H

I (x2min ) = 0. x^2

;

dI = d + F ( + ) ; d ; F ( ;) = F ( + ) ; F ( ; ) = dx^2 dx^2 dx^2 A2 A2 + ; 2 2 2 3 2 3 = 12 sin (x1 ) ;+ cos x2^2 + (r ) ; sin (x1 ) ;; cos x2^2 + (r ) : ;x1 + (r ) ;x1 + (r ) dI = D &

A2

F =X(r2), , x^ !lim 2 x2min dx^2 dI = +1. x^2 = 0

dI = 1 sin (x; ; x+ + (r2 )) = = x^ !lim 1 1 dx^2 2 2 x2max dx^2 dI = ; X(r) < 0. dx^2 & I (^x2 )

(x2min 0) , (0 x2max) | . 6 ;

ddIx^2 = AF2 ( + ) ; AF2 ( ; ) = 0.

0/ 2 x^ (28) ; tg = x^;2 +2 + (r): x1 x1 2 I (^x2 ) Imin . 0 , Imin , (

. . Imin = ;X(r2 ) 1 . I (^x2 ) = Imin = ;X(r2 ). W l, H 0 2 , 2 , (F + uG) dt < 0. C ! l x_ = A + B , /(! ( + ). ; u = 1 X(r) ;

, 1 ! (Vx1 xV2). 6 x_ 2 = (r)

xV2 = x^2 + (r2 ), , xV1 = x;1 + (r2 ). 6 (Vx1 xV2) ! u = 0 1

( + ). H

; 4

I

(F + uG) dt =

('xZ1 x'2 )

(xZ1 x^2) ;

(F + G) dt +

.

(x+1 x^2 ) F dx + Z F dt = A2 2

('x1 x'2 ) (x1 x^2) 2 2 = X(r)(r ) + (r )(r) + Imin = ;X(r2 ) < 0: 2 2 4 Imin = +X(r ), ! -

(x+1 x^2)

15.

,

;

4 . I (^x2 ) = Imin = +X(r2 ). 0 , ( 0 , 2 = 2 (0 0 ) > 0 .

.

231

. , ( + ) ( ; ) 0

x_ = A. ; ; / x~

(x;1 )2 + x^22 = X((x+1 )2 + x^22 ). : 2 ; (28), ; xx^+12 = X(1). ; ( + ) 0 ! 0 K0 !, 1 ! ; . 0 p0 = ( + ) , 20 P ( + ). H,

x1 (p0) = x+1 , x2(p0 ) = x^2. H (26) x2(p1 ) = x^2 + (r2 ), (27) (28) | x1(p1 ) = x;1 + o(r). 5 2 , 0 p1, ( ; )

: d(p1 ( ; )) = o(r). o() / s ! x~, s &

A2 x~. C

(t) (t) x_ = A, /( t = ; p1 ( ; ) . :;&&

/ ; , ; (t) (t) t 2 ( ; + ) 1 (1 ! , !(! o(r). 5 2 ,

F , / : R+ R+ F ((t)) dt = F ((t)) dt + o(r2 ) = Imin + o(r2 ) = X(r2 ). : ,

0 = AF2 (( + )) ; AF2 (( ; )) = AF2 (( + )) ; AF2 (p1) + o(r). D A2 = ;dAB + (r2 ), ; dFxAB2 (( + )) ; dFxAB2 (p1 ) = o(r2 ). ; (17), !(

& ! H1(t)

, H1( + ) = Z + 1 Fx 2 + + = ;A x + A x ;F x^2( ) + dAB ( ) ; 2F dt + d (p1 ) + (r2 ) = 1 2 2 1 AB = ; X(1r2 ) f;2dAB (( + ))(X(r2 ) + o(r2 ))g + (r2 ) = ;X(r): ! ( + ) . . 2 = ( tV) H1(2 ) = 0. H

H1( + ) = R2 H_ 1 R2 (r) + =; d = ; _ &(1) d = ; X(r). ! 2 = ( ) + X(1). ( + ) ( + ) C 0 p0 = ( + ) ( + )

o(r), ; ( + ) = 0 + o(1). ! 2 = 0 + X(1). D , 0 2 s ! x~ 0 , 2 . , 2 = 2(0 0 ) > 0 . 5 2 , ( ! 0 1 , 0 2 2(0 0 ) = 0 , 1 0 < 0 < 0 2 ( . . 5). 5 (0 01 ) (0 02 ) | 0 20 P. .& 0 = 01 . C dAB2 (dAB0 01 ) & ! dAB0 . H/ 01 < 0 , , ;

;

;

232

.

( c < 1, ;c dAB0 > ;dAB2 . 0 = 02 ;dAB0 < ; dAB2 , c < 1. d2AB0 @dAB2 = @2 = 6 (0 0 1)

@ > c12 > 1, @d lim d 0 AB0 dAB0 !0 2AB2 @d d AB2 6 c < 1, AB2 = 0. 6 (0 2 ) | @2 < 1, = d lim!0 dAB 0 @0 @0 0 AB0 @dAB2 > 1. !, , , (0 ) =

2 0 0 @dAB0 1 2

, 0 0 . 5 2 , 0 (0 0 1) 2 . 2 / 2 / 2 0 , ., , 12]. ) f0g ]. D 0 & = (dAB ) C 1 . 5 2 , 2 && ! !, ! 20 P !(! ! 0 1. 2 ; ! 01, 1 M; | ^01. G p^0 2 ! ^01 , p^1 2 ! ! ^10 , p^2 | ^01. 10 ^10 ! 1 0. 5

^ (18), (19), !( ^01 ^10 , 2! T X

2 (2), (8). 0 PC 1 / / ^01 ^10 . D , , ; 2 ! (x = 0 = 0), N T X . 5 x~ !- ;

. B 0 N &

F ! 0 13. 5 2 , , N , . , 4, !. ) 0 15 . 2 B 2 4 !(

0 . 2 ( ~( )) . 2 (~( ) 23 ] Imin = ;X(r2 ) 2 Imin = +X(r ) B Imin . 6 0 . 5 2 ,

2 (0 0) = 0

0 , !( 1. 5 . 4. . 02 , x~ - . 4

.

,

. 4

,

233

Y 2 & 4. H. . (, ; 2.

+

1] . . . | .: , 1975. 2] #$ %. . 1 -' ( ) ( ) *( '+ '( ,- ,**' // / - 0 $ ,1. | 1977. | 2. 11, 3 1. | 4. 57{58. 3] 8 . ., 8 9. /., : %. *),'' ,') $ *'0 $ ** ,') ( 0 + // 2 . | 2000. 4] ='$ 9. 4. . | .: >*. 0 . 0.-)'. '., 1961. 5] ='$ 9. 4., #'*- . >., >) 0 %. ., @ A. /. ')'+* ' ,') ( ,**. | .: , 1969. 6] Davydov A. A. Qualitative theory of Control Systems. | Providence, RI, 1991. | Translations of Mathematical Monographs, vol. 141. 7] Fuller A. T. Constant-ratio trajectories in optimal control systems // Internat. J. Control. | 1993. | Vol. 58, no. 6. | P. 1409{1435. 8] Jakubczyk B., Respondek W. Feedback classiGcation of analytic control systems in the plane // Analysis of controlled dynamical systems (Lyon, 1990). | P. 263{273N Progr. Systems Control Theory, vol. 8. | Boston: BirkhOauser Boston, 1991. 9] Kelley H. J. A second variation test for singular extremals // AIAA J. 2. | 1964. | No. 8. | P. 1380{1382. 10] Kelley H. J., Kopp R. E., Moyer M. G. Singular extremals // Topics in optimization. | N.Y.: Acad. Press, 1967. | P. 63{101. 11] Krener A. J. Approximate linearization by state feedback and coordinate change // Systems Control Lett. | 1984. | Vol. 5, no. 3. | P. 181{185. 12] Nitecki Z. Di^erentiable dynamics. | Cambridge: M.I.T. Press, 1971. 13] Zelikin M. I. On the singular arcs // Problems of Control and Information Theory. | 1985. | Vol. 14, no. 2. 14] Zelikin M. I., Borisov V. F. Theory of Chattering Control with Applications to Astronautics, Robotics, Economics and Engineering. | Boston: BirkhOauser, 1994. ' ( 2000 .

. .

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Abstract A. A. Chilikov, Taylor power series of algebraic functions over elds of positive characteristics, Fundamentalnaya i prikladnaya matematika, vol. 7 (2001), no. 1, pp. 235{256. In this work we show algorithmical solvability for the problem of calculation of the Taylor power series for an algebraic function over a 5eld of positive characteristics. An e6cient algorithm for construction of a 5nite automaton solving this problem is given.

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. . | .: , 1986. 4] . &'. '( . | .: , 1968. 5] *. + (,. p- ./ / , p- ./0 . -,. | .: , 1982. 6] 1. 2. /. 3/ 4 ( 5( . | .: * , 1993. 7] A. J. Belov, V. V. Borisenko, V. N. Latyshev. Monomial algebras. | NY: Plenum. ' ( ) 2000 .

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w

x

x

w

w

Abstract V. V. Shchigolev, On leading monomials of some T-ideals, Fundamentalnaya i prikladnaya matematika, vol. 7 (2001), no. 1, pp. 257{266.

In this paper some analogs of the Gr'obner base for T-ideals are considered. A sequence of normal monomials of the T-ideal 2(3) is built so that the monomials are independent w.r.t. the operation of monotonous substitution and the insertion operation. Also a theorem is proved stating that for algebras without 1 a multilinear identity of the form 1 , 1 2 ] 2 , where 1 , 2 are variables and 1 , 2 are monomials, belongs to every T-ideal that is 8nitely based w.r.t. the inclusion relation of the leading monomials. T

w

x x

w

x

x

w

w

x 1.

1] T- . ". #. $ 2] & '

. ( ) * & , *, . ( - * * * * , & ,* . * T- . / * (. 0. 1 . 2 & K | , x1 : : : xn : : : | . ( ' ), *: xi < xj , 2001, 7, 9 1, . 257{266. c 2001 !, "# $% &

258

. .

i < j. 7 ' * - , & * , ) * - . 2 & S | x1 : : : xn : : :, , S (n) | , *, n, S0(n) | x(1) : : :x(n) , 2 Sn , Sn | *

n. (2 9 ' S.) 1 K, * ' S0(n) , P (n). ( ) jwj w , & w , ** * w. 7 u , u = u1xi u2xj u3 , i < j. < S0n x1 : : :xn . 2 x 2 v, x | v. =* ) c() ), , & '

, x1 : : : xn : : :, * ) i degxi = degxi c(). ( * S ), : xi1 : : :xin xj1 : : :xjm & , & & i1 : : : in ** * & &) & j1 : : : jm . > v u, , v * u (u v). 2 , * S, , v u, v | u. 0 S ' ), : xi1 : : :xin xj1 : : :xjn , t fi1 : : : ing fj1 : : : jng, t(ik ) = jk , k = 1 : : : n, . ? t - F1 F2 F, fxi1 : : : xin g fxj1 : : : xjn g . > f1 2 F1 , f2 2 F2 f1

f2 ' -, , f1 f2 . 2 & F = K hx1 : : : xn : : :i | * * 1. =* f 2 F f@ f, * ) I F I@ = ff@: f 2 I g. 7 , , I@, & I. = , S(I) = S \ I@. 2* & S= ), : * ' B1 B2 B1 B2 ( B1 B2 ), , ) u1 2 B1 u2 2 B2 , u1 u2 ( u1 u2). 2 & (A 6) | * . / a 2 A (A 6)- & fa1 : : : ang A, , ai, ai 6 a. < (A 6) , , ) a1 : : : an 2 A, ) ' a 2 A (A 6)- & fa1 : : : ang.

259

T-

( ) * * (S(I)= ) (S(I)= ) , I ** * T- . 7 u 2 S (T- ) & ff1 : : : fk g F , u ** * (S )- ((S )- ) & ff@1 : : : f@k g. B Tk(3) T- F, x1 x2 x3] : : :x3k;2 x3k;1 x3k ]. 2 & T (3) = T1(3) . ? ; S, ) xk xj xi xk xi xj (1) (3) i < j < k. 1 &, S n S(T ) ;. ( ), & : , & & w1 : : :wn : : : S n S(T2(3) ), * i 6= j wi wj - ) D & I | T- F ( 1), * (S(I)= ) . E I

w1 x1 x2]w2, w1 w2 | .

x 2.

1 &, ) u 2 ; * * r Q u = ui , ) ui vi xki , xki & ) i=1 vi vi , i < j ) * ui & ) uj , ui , juij = 1. 2 & & (u1 : : : ur ) u. 1. ; T (3). . 2 . E w = r Q = ui 2 ; ** * f 2 T (3) , i=1 (u1 : : : ur ) | . 2 & ui = vi xli . > ui , vi , , . > ui , ui . 2 ) * i = 1 : : : r. E * w0 f 0 ,

), * f . = &, & w0 < w w0 | f, w00 | , w0 . 2 & w0 = xl 0 , w = xm , tQ ;1 l < m. ? ) , , t, = ui u0t , i=1

260

. .

Q r

= u00t ui , u0txm u00t = ut . H * * i=t+1 , xl ut+1 : : : ur . 7 , xl . 7 &, xl u00t . 2' ), *: ju00t j > 2 ju00t j = 1. 2 & * .E u00t , & xm , & ** u00t . ? ) u00t = vt00 xsxl , s > m. 2 xm u00t xsxl , xl 0 | xl 00 . ? ) w00 < w0 . 2 & * . E u00t = xl , xm u00t *, xl 0 xl 00 . ? ) w00 < w0 . E T (3) , & & , f 0 2 T (3). 7 w0 s Q w0 = (xli xmi ), xli > xmi (xl1 xm1 : : : xls xms ) | i=1 . r Q > & v 2 P (n) | & , v = vi , i=1 (v1 : : :vr ) | v. > &, , ) i1 < : : : < is , jvik j > 2 k = 1 : : : s, w0 v v & & . 2' &, , ) 1 6 i1 < : : : < is;1 6 r, i 2= fi1 : : : is;1g jvi j = 1. = , , P (n) * jv1j : : : jvr j. H &* ' - , & & P (n). ?, r 6 n. =* i1 < : : : < is;1 * Cns;1 . ? * & & * jvi1 j : : : jvis;1 j. E jvi j 6 n * i = 1 : : : r, ' ns;1. ( Cns;1ns;1 & P (n). / , dn = dim(P (n) j P (n) \ T (3) ) T (3) , , , , 3]. 1 . E , 1. ; = S n S(T (3) ). ( * T (3) (. 0. 1 4]. 2 & I1 , I2 | F. E F I2 + +I1 F | F F. 7 F F

* u v, u v | F . 7 , u2 v2 > u1 v1, u2 > u1 u2 = u1 v2 > v1. =* f 2 F F f@ f. 0 u v & & L F F , * ) f 2 L u v 6= f.@ 7 ), * 2. u v 2 F F , u v | I1 I2 . u v | F I2 + I1 F .

T-

261

. 2 . E , f 2 2 F I2 + I1 F, f@ = u v. 2 f ), : f=

n X i=1

u i fi +

m X j =1

gj vj

(2)

* ) i j 0 6= fi 2 I2 , 0 6= gj 2 I1 , ui vj | , ui > uj vi > vj i > j. u. 7 ui f@i & (2). E f@ > ui f@i > u v, . 2' * i = 1 : : : n ui 6 u. 2 & ui0 = u, ui0 f@i0 & (2). 2' f@i0 6 v. E v &, f@i0 < v fi0 & v. 7 &, u v (2). 2 & & vj0 = v. > g@j0 > u, , , , i, ui = g@j0 . ( ' g@j0 v

& (2). E f@ > g@j0 v > u v, . 2' g@j0 6 u, u & & I1 , g@j0 < u gj0 & u. ? ) , u v (2) , (2) | . 1 . . H &, u1 : : : un 2 F : : : F & & I1 F : : :F +F I2 F : : :F +: : : + + F : : : F In & , u1 : : : un & & I1 : : : In , w1 : : : wn > v1 : : : vn * , , i = 1 : : : n, wl = vl l < i wi > vi . 2 ' E- . 2 & w = uv 2 P (n). ( & Iuv , * ), ), : 1) yx, y 2 v x 2 u, 2) u1u2 u3 u4]u5v1, c(u1u2 u3u4u5 ) = c(u), c(v1 ) = c(v), 3) u1v1 v2 v3 v4 ]v5, c(v1 v2 v3 v4v5 ) = c(v), c(u1 ) = c(u). 3. T2(3) \ P (n) Iuv \ P (n).

262

. .

. 2 &

f = tt1 t2 t3]r1 r2 r3]r

tt1t2 t3 r1r2r3 r 2 P (n). 2 & w1 = t, w2 = t1t2 t3 , w3 = r1r2r3 , w4 = r. > , )

i j = 1 : : : 4, i < j wi v, wj

u, f 2 Iuv 1). 7 &, &, , l = 1 : : : 4, i < l c(wi ) | c(u) i > l c(wi ) | c(v). ( &, l > 3. Em P f = i tt1 t2 t3]i , i | r1 r2 r3 r i=1

*. > i ' 1), i = 0i 00i , c(0i ) c(u) (00i ) = c(v). ? ) t11 t2 t3]i 2 Iuv . 1 . 4. u v | T (3) w = uv 2 ( n 2 P ). w | Iuv , T2(3) ,

3. . 2 , & w = f,@ f 2 Iuv \ P (n). 2 & V0 | K, * P (n), , 1), V1 | , * , , 1). E P (n) = V0 V1 . L , Iuv \ P (n) & ' . E w 2 V1 , w = f@0 , f0 f V0 . H f0 2 Iuv . 2 f0 f0 =

n X i=1

uifi +

m X j =1

gj vj

* ) i j c(ui) = c(u), c(vj ) = c(v), ui fi | 3), gj vj |

2) * ) 1 2 fi gj c(1 ) = c(v) c(2 ) = c(u). M , fi 2 T (3) gj 2 T (3) . E u v | n

m

i=1

j =1

X X f 0 = ui fi + gj vj

F T (3) + T (3) F . 2 2 * u v | T (3), ). 1 . ? ) 4. u v 2 ; uv 2 S . uv | (3) T- T2 . 1. w1 : : :wn : : :

S n S(T2(3) ), i 6= j wi wj - .

263

T-

. B ;0 w 2 S, ),

w = uv, u v 2 ;. 2 & D |

w = uv 2 ;0, u v 2 ;. B wD , uD vD , w, u v xi ! 1, xi 2 D. 1 &, w0 = u0v0 , u0 v0 2 ;, w0 2 ;0 w0 w (mod ), u0 u (mod ) v0 v (mod ). = &, w0 w (mod ), * D w0 wD . 2 , , ) u0 v0 , u0 uD , v0 vD u0 v0 = w0 . E u0 v0 2 ;,

u0 = u0 v0 = v0 . ? ) u0 uD , v0 vD u0 u (mod ), v0 v (mod ). H * ' - * 4, * &,

& & wn =

nY ;1 k=0

x4k+6x4k+3 x4n+5x4n+6x4n+3x2x4 x1

** * . E .

nY ;1 k=0

x4k+8x4k+5 n 2 N

x 3.

# & * T- I , * * (S(I)= ) . ( NH & 0 { ( . 5]). 2. I | ! T- ! F 1, ! " (S(I)= ) . I " !! w1x1 x2]w2, w1 w2 | . . ( ), , & * ' & : x, y, z, , & . 2 & 0 6= f 2 I \ P (d) . 2 ) , ) f1 : : : fN 2 F, * ) g 2 I * i g0, g0 fi g@0 g@. (3) 0 2 & m = maxfdeg fi : i = 1 : : : N g. 7 f , ), * f xk ! wk = xk x(k;1)m+d+1 : : :xkm+d , k = 1 : : : d, ** * w1 : : :wd . H (3) , , S(I) m : 1) u, 2) uyv, uv , y & uv. ( xm 2 I F=I & NH.

264

. .

# . H h(x y 1 : : : n1 1 : : : n2 ) = uyv ;

n1 X i=0

uiyhi 2 I \ P (m)

(4)

1 : : : n1 x = u, 1 : : : n2 = v, juij = i 6 n1 = juj ; 1 ui | u. 2 un1+1 = u uyv h(x y 1 : : : n1 1 : : : n2 ). ? r & i = 0 : : : n1, '-- hi 0, , ) , ;1 . # r = ;1. M x1 : : : xm x1. E (4) * '-- hi xm1 2 I, NH * & & & F=I. # & , r . H jur+1 j 6 juj. 2 ur+1 ur+1 = ur xt. M & xt 6= y. B h0 h0i &

xt ! xm+1 xt h hi , i = 0 : : : r. > r = 0, h00 = xm+1 h h00i = xm+1 hi , r > 1, h00 h00i & r ! r xm+1 h hi , i = 1 : : : r ; 1. rP ;1 H h00 ; h0 = ur (xm+1 yhr ; yh0r ) + uiy(h00i ; h0i ). 2* i=0 x1 : : : xm+1 , y, x, y, ),

I:

xr+1 yxs ; xr yxs+1 s = n1 + n2 ; r + 1 6= 0, '-- hr . 2 i < r h0i h00i &, y. > 6= , F=I & NH. > = , (5) xr y x]xs 2 I jr + sj 6 m ; 1. H (5) , xp y x]xq 2 I, p = m + 2r, q = m + r + 2s. E char K = 0, (x + z)p y x + z](x + z)q

I. ( , pX ;1 qX ;1 xp;i;1zxi y x]xq + xpy x]xj zxq;j ;1 + xpy z]xq 2 I: i=0 j =0

(6)

2 , (6) I. 2 i > r xp;i;1zxiy x]xq 2 I - (5). 2 & & i < r. . g1 = h(zxi y x]xs x : : : x) g10 = = xn1 zxi y x]xs+n2 . E i < r, p ; n1 ; i ; 1 > r, &, xp;n1;i;1 (g1 ; g10 )xq;s;n2 2 I. ? ) xp;n1;i;1 g10 xq;s;n2 =xp;i;1zxi y x]xq 2 2 I.

T-

265

" , j > s, xp y x]xj zxq;j ;1 2 I - (5). 2 &

& j < s. . g2 = h(y x] xj zxr x : : : x) g20 = xn1 y x]xj zxr+n2 . E j < s, q ; j ; 1 ; r ; n2 > s xp;n1 (g2 ; g20 )xq;j ;1;r;n2 2 I. ? ) xp;n1g20 xq;j ;1;r;n2 =xpy x]xj zxq;j ;1 2 2 I. E , (6) I, & xp z y]xq 2 I: (7) s 0 n n + 1 2 . g3 =h( z y]x x : : : x) g3 =x z y]x s , , &, p ; n1 > 2r > r, xp;n1 (g3 ; g30 )xq;n2 ;s 2 I: (8) p ; n 0 q ; n ; s p q 2' x 1 g3x 2 = x z y]x 2 I. H * xp 1 z y] 2 xq 2 I: (9) B F0 F 1, ) - . 2 & & f 2 F, , (7), (8), (9), f0p+q+1 2 I, f0 | & ' F0

f. H NH , , t, F0t I , , x1 x2]x3x4 x5]x6 : : :x3(t;1)x3(t;1)+1 x3(t;1)+2] 2 I. 2 & wk = wk0 x(k+1)(m+1) xm+k(m+1) * k = 0 : : : t ; 1, wk0 = = x1+k(m+1) : : : xm;1+k(m+1) . E w0 : : :wt;1 ** * tQ ;1

wk0 x(k+1)(m+1) xm+k(m+1) ] I. k=0 2 (3) , S(I) m : 1) , 2) uxyv, uxv uyv y & x. ( F=I & NH. ( uxyv = t0 P = iuyvi , c(uxyv) = c(uyvi ) * i. 2 ** x 0 i=1 t P v, uxyxk = uyxk+1, = i. > 6= 1, * & i=1 F=I & NH. > = 1, ux y]xk 2 I: (10) ( ux + z y](x + z)3k I. ( , 3X k ;1 ux y]xizx3k;i;1 + uy z]x3k: (11) i=0

> i > k, ux y]xizx3k;i;1 2 I (10). > i < k, I 3 ux y]xiz xk]x2k;i;1 = ux y]xizx3k;i;1 ; ux y]xi+kzx2k;i;1, (10) ux y]xizx3k;i;1 2 I, 2k ; i ; 1 > k. H (11) uy z]x3k 2 I. 0 M *, uy z] x3k 2 I, ** NH, u1y z]u2 2 I \ P (s ) * & s0 . E .

266

. .

M , (S(I)= ) , F=I , (4). = * ,*, jw1j jw2j > 1. = &, & I | T- F , x1 : : :xk xk+1 xk+2]xk+3 : : :xk+3+l x1 x2]x3 x4], k l > 1. 2 , (S(I)= ) m | & * & ' . 7 x2x1x3 : : :xm xm+2 xm+1 ** * x2 x1]x3 : : :xm xm+2 xm+1] I. ( * , ** * I, x2x1x3 : : :xm x3 : : :xm xm+2 xm+1 | I. E n xn 2 I, * * x y]xm;2 = 0 xm;2 x y] = 0. H * . ( W K, * fxiyxm;i;1 : i = 0 : : : m ; 1g. 2 W0 W , * fxiyxm;i;1 : i = 1 : : : m ; 1g. 1 &, u1u2 u3]u4 2 W0 u1 u2]u3 u4] 2 W0, & u1u2 u3u4 2 W , ju1j > k, ju4j > l u1 u2u3 u4 2 W . 7 , &, x y]xm;2 2= W0 . / , x y]xm;2 2= I, ) (S(I)= ). " (. H. 1 *.

!

1] W. Specht. Gesetze in Ringen // Math. Z. | 1950. | Vol. 52, no. 5. | P. 557{589. 2] . . . !"# $#%& '( // '( '(). | 1987. | * 5. | +. 597{641. 3] . . . /% (, ( 0% ) 0!"% PI-'(%. | 2.. . . )". 3.-. ). | /#), 1981. 4] 4. /. 5%6#. 7 #% % # " T-"' // +89. | 1963. | :. 4, * 5. | +. 1122{1127. 5] G. Higman. On a conjecture of Nagata // Proc. Cambrige Philos. Soc. | 1956. | Vol. 52, no. 1. | P. 1{4. ' ( ) 1998 .

; y v ; z w + z + x = 4t

; x u y

. .

. . . 511.3

: , .

,

$ $ . x y z t u v w

; x u

y

+

; y v

z

+

; z w

x

= 4 ! "t

Abstract M. Z. Garaev, On the diophantine equation ; xy u + ; zy v + ; xz w = 4 , Funda-

mentalnaya i prikladnaya matematika, vol. 7 (2001), no. 1, pp. 267{270. It is proved that the equation positive integers . x y z t u v w

x

; x u

y

+

; y v

z

+

; z w

x

t

= 4 has no solutions in t

1.

1] , n n2 +3n+9 n ; 3 3k + 2, x+ y + z =n (1) y z x

$% & & x y z. ( )% (1) $ n = ;6 (. (. +,, n = 1 | (.. . +. / & n 2 f;1 5g | 2. (. 3. 2] , $ (1) & n, 4& 3. 3] (1) ,& & x y z. 5 , n 2 f4k 8k ; 1 22m+1(2k ; 1) + 3g, k m , , (1) $% ,& & x y z. 8 , 4% 4 ..

, 2001, 7, , 1, . 267{270. c 2001 , !" #$ %

268

. .

1.

xu + yv + z w = 4t xyz (2) x y z t u v w (x y z) = 1. . n1 n2 n3 | , n = n1 + n2 + n3, xn +yn +z n = 4t xn1 yn2 z n3 x y z t. 2. x u + y v + z w = 4t (3) y z x

x y z t u v w. x

2. 1

( , 9 : ;. < , , x y z t u v w , (2) (x y z) = 1. . , a) uvw 9 b) 9. 8 % a) uvw | 9 . = 4 . ,, u | 9 . > xu + yv 0 (mod z(4t xy ; z w;1 )) , v | 9 . @, w | . 9 . / , y, z | 9 . < ,, , z | 9 . A y = 2k y1 , k | , , y1 | 9 , . const n;1 0) j i ( )j 6 const 8 2 !0 ] = 1 1 T

P

n

v

L

Tv

x

T

v

v v

P

T

u

T

u u

p x

x

L

P

u u dx

P

T

v

x

i

x

i

>

:

, 2001, 7, / 1, . 271{274. c 2001 , !" #$ %

P u

u

272

. .

0 ( N X

i (x)ui(y) ;

N X

u

i=1

i (x)vi (y):

v

i=1

0 , -

X i ( ) i( ) u

i 6

x u

x

(1), (1)

, T ( ) T +P ( ) + . 0

, o

K

T

T

K

O

x y

K

x y

P

T +P (x y )

=

T (x y ) ;

Z1

K

K

T (x z )p(z )KT +P (z y ) dz

(1)

0

K

T (x y )

T +P (

K

=

1 v (x)v (y) X i i

)=

x y

i;

i=1

1 u (x)u (y) X i i i=1

i;

, ( 1 (0 1). & (1) ;1 N = f j j j = ( N + N +1 )2 g

( (1) , ;N = f j Re = ( N + N +1 )2;1g ( , (1) j j;2) , L

K

N X

i(x)ui (y)

u

i=1

=

N X

1 Z i( ) i( ) + p 2 ;1

v

i=1

x v

y

Z1 K

T (x z )p(z )KT +P (z y ) dz d:

;N 0

5, 6 (. !2]), , 1 1 Z Z p KT (x z )p(z )KT +P (z y ) d dz = 2 ;1 p 0 ;N Z1 Z p 1=p 1 = 2 KT (x z )p(z )KT +P (z y ) d dz 6 0 ;N

273

6 21

Z Z1

j T( K

;N

T +P (

) jp

1=p

z y dz

0

Z

d

1=p

k(T ; E );1 kp k(T + P ; E );1 kp d

6 const

1

;N Z1

6 const 1

)()

x z p z K

;1

(j j +

d

1=p

(1)

;1)2"p

o

n;1)(1;")2p;p

N(

Ci

6

6

= ( ;n+2 ) o N

= Im

0, . 1. 02 ; 4 1 ;1 6= 0 > 2 > " >

p. 1

1) 2)

N X

u

i=1

n

i (x)ui (y)

=

i (x)ui (y)

=

u

i=1 N X

N X v

i (x)vi (y) + o(N

v

i (x)vi (y) +

i=1 N X i=1

;n+2))

+ 1 ( ) + 2 ( ) + + k ( ) + ( ;(n;1)(k+1)+1) 1( ) 2( ) k( ) | , (1) . 8 !6] 2.

N

N

N : : :

N X

i=1

N

:::

N

o N

N

i=

N X

i+

N X

i=1

(

i

i) + o(N

pv v

i=1

;n+2 ):

3. & - !6]. 4. : !3{5] 6 !2] .

1] . . . | .: , 1969. 2] ! ". "., # $. %., & ". #'#. | .: "(( , 1948. 3] $,#' -. -. ' ' , #. .

// $ #1. | 1990. | 3. 26,

4 3. | 5. 533{534.

274

. .

lp M

4] $,#' -. -. 8 ' ' 9 # # 5. 69{75.

(

) // $ #1. | 1992. | 3. 28, 4 1. |

5] $,#' -. -., &=# . 5. 8 ' ' ''1>9 '#9 # // $ #1. | 1993. | 3. 29, 4 5. | 5. 852{858. 6] "'# . ". ? ' @' ','#9 @= #9 ''1>9 # // $ 555A. | 1963. | 3. 150, 4 6. | 5. 1202{1205.

& ' 1997 .

0 2 ] L2

. .

. . .

517.51

: , .

, ! " $!" 2 %0 2 ]. L

Abstract

M. G. Esmaganbetov, Minimization of exact constants in Jackson type inequalities and diameters of functions belonging to L2 %0 2], Fundamentalnaya i prikladnaya matematika, vol. 7 (2001), no. 1, pp. 275{280.

We obtain a series of results related to minimization of exact constans in Jackson type inequalities as well as the diameters of functions belonging to 2 %0 2 ]. L

L2 | 2- 12 Z2 1 2 kf k2 = jf(x)j dx 0

" # 1 X f k cos(kx + ') (0 0): k=1

& L2 ' , ) " *+" f () 2 L2 (L02 L2 f 0 f): ,+ Sn;1 (f () - x) | ( > 0)- " " n ; 1 " #

f () (x), , . ,

+. + '

f () 2 L2 + Tn . n ; 1 kf () (x) ; Tn (x)k = En;1(f () ) = E(f () - K2Tn;1) = inf Tn 21 1 X ( ) ( ) 2 2 = kf (x) ; Sn;1 (f - x)k = k k (1) k=n

, 2001, 7, - 1, . 275{280. c 2001 !"#, $% &' (

276

. .

K2Tn;1 | (2n ; 1)- + L2 . N- 2 + , + + H L2 3 +3) + : dN dN (H L2 ) = inf sup inf kf ; uk KN f 2H u2KN

N N (H L2) = inf

inf

sup kf ; Af k

N N (H L2) = inf

inf

sup kf ; Af k

KN A2L(L2 KN ) f 2H

KN A2L1 (L2 KN ) f 2H

L(L2 KN ) | ' + L2 N- KN , L1 (L2 KN ) | ' L(L2 KN ), . . + A + " KN , Af = f +" f 2 KN . X 1 r r k !r (f- ) = sup k5h f(x)k = sup (;1) k f(x + (r ; k)h) | 06h6 06h6 k=0 + + " r > 0

f(x). 2 2 ( ) r & Wr (f - ) !rr (f () - t) 60 ] cos 2t , +""

0 R 1r ( ) r B 0 !r (f - t) cos 2 t dt CC Wr (f () - ) = B B@ R CA : cos t dt 2

2

0

2

Hr(!( )) | ' f 2 L(2) , Wr (f () - ) 6 !( ) 8. 9. & 61] + + 3 ' En;1(f) !1(f () - t) 60 n ] sin nt. : ) .+ ; ' . *. < 62,3], 8. >++ 64], *. *. @ + 65], B. C. , 66] . c(r) | + ' + " , ") " r. < c(r)!r (f () - ) 6 Wr (f () - ) 6 !r (f () - ) + Hr() + " +

HD r(!) = f!r (f () - ) 6 !( )g + " + . + Hr(!), E Wr (f () - ) , - , + +"

+. + '

.

277

...

F +

+ HD r(!) + G. H. G '

+ ' 1975 . . 2 +. + E 62{6] + + + " ' !( ) + + + +" . 8 + " + Hr (!( )) + + + ' !( ). 2 , +" 8. . 2 67], . +

+ KNT ) XrN (L2 L2 KN ) = sup WE(f (2) f 2L2 f 6=const r (f () - ) N, . . + + " + (2) KN L2 N (8r > 1, > 0, N = 1 2 3 : : :): KNT ) XrN (L2 L2) = inf sup WE(f (3) ( KN f 2L2 f 6=const r (f ) - ) , + (3) 3 + Hr (!( )) 0 < !( ) < 1. : , r = 1, = 0 1 2 : ::, 0 < 6 n , N = 2n ; 1, n = 1 2 : : : (2) . 64]. 1. r > 1, > 0, 0 < 6 n , N = 2n ; 1 N = 2n ( sin n )2 ; 2( n)2 r2 1 !( ): (4) dN = N = N = n 4 2 2; 4( n)2 Sn;1 (f- x).

. < r > 1, , ""

Zb X n a

0 < p 6 1,

Z X 1 0

>

k=n

1 X

k=n

k=m

p p X n Zb

juk (x)j dx

1

>

k=m a

juk jp dx

2r r

2 sin kt2 k22k cos t dt 2 >

k22k

p 1

1

Z 0

2 r X Z r 1 2 sin kt2 cos 2

t dt = k22k 2(1 ; cos kt) cos 2 t dt : k=n

K +, 64], 0 < 6 n "

Z t dt '(y) = cos yt cos 2

0

0

278

. .

y. E +

2r kh 2 sin 2 k22k

1

X k5rh f () k2 = k=1

,

Z

(5)

r X Z r 1 2 2 ! (f - t) cos 2 t dt > k k 2(1 cos nt) cos 2 t dt : k=n 0 0 :3 (1) + 2

rr

()

3r 2 R ( ) r 6 !r (f - t) cos 2 t dt 77 kf ; Sn;1(f)k 6 n1 664 0R ; 75 2 nt cos t dt 2 sin 2

2

0

2

; +"" +" f(x) = cos nx: < Sn;1 (Tn;1(x)) = Tn;1(x) +" Tn;1 (x) 2 KnT;1 Sn;1 (f- x) 2 K2Tn;1 +" +3 f 2 L2 , +" + Hr(!( ))

2 R 3r 66 0 cos 2 t dt 77 !( ) 2n 6 2n;1 6 sup kf ; Sn;1(f)k 6 n 64 R ; 75 : (6) f 2Lr 2 sin nt2 2 cos 2 t dt 2

2

0

* + dN 6 N 6 N (6) + +" :

2 R cos t dt 3 r 77 66 2 0 dN 6 N 6 N 6 !( ) 75 : 6 n 4 R ; nt 2 2 sin 2 cos 2 t dt 2

(7)

0

& , .

2 R 3r 66 0 cos 2 t dt 77 T R = !( ) 75 U2n+1 = n 64 R ; nt 2 2 sin cos 2 t dt 0 8 2 R 3r 9 > > > > t dt cos n < = 6 7 2 X !( ) 6 7 0 = >Tn(x) = ck cos(kx + 'k ): kTn k 6 n 64 R ; 7 5> k=1 > 2 sin nt 2 cos 2 t dt > : 2

2

0

279

...

(2n + 1)- K2Tn+1 - + . ', Tn 2 Hr (!( )). * + (7) 0 6 kt2 6 nt2 6 2

2 R 3r () r 66 0 !r (Tn - t) cos 2 t dt 77 64 R cos t dt 75 = 2 3r 2 0 Pn ; r R 2 r 2 sin kh2 k2c2k cos 2 t dt 77 66 06sup h 6 t k =1 0 77 6 = 66 R cos d dt 4 5 2 0 2 R ; 3r 2 nt r 6 n 2 sin 2 cos 2 t dt 77 6 664 0 R 75 kTnk 6 !( ): cos d dt 2

2

1

2

2

2

0

2

E R Hr(!( )). < . (. 67, . 347]) + 3 r2 2 R 66 0 2 cos 2 t dt 77 !( ) d2n;1 > d2n > n 64 R ; 75 : 2 sin nt cos 2 d dt 0

:3 + (7) + 1. 2. > 0, r > 0, N = 1 2 3 : ::, 0 < ! 6 1. XrN (L2 L2) = dN (Hr(!( ))- L2 ): (8) ( )

. L2 Wr (f - ) = u > 0. ,+ f1 (x) = () ; 1 = u !( )f(x), Wr (f1 - ) = !( ), . . f1 2 Hr (!( )). M " + ' +3 + E(f- KN )

Wr (f () - ) KN ) 6 sup E(f- K ): sup WE(fN ( (f f 2L2 r ) - ) f 2Hr " ' " KN L2 N, + Xrn (L2 L2 ) 6 dN (Hr (!( ))- L2 ): (9)

280

. .

G , +" +3

f 2 Hr(!( )) + + " +

Hr(!( )) KN ) E(f- KN ) 6 WE(f( (f ) - ) r

E +" ' KN , 3 + , (9). 1, > 0, 0 < 6 n , 0 6 !( ) 6 1, N = 2n ; 1 N = 2n ; n 2 ; 2( n)2 ; r2 XrN (L2 L2) = n1 4 sin2 2; 4( n)2 !( ):

1] . . " L2 0 2] // &. '. | 1967. | ,. 2, . 5. | 0. 513{522. 2] ," 3. 4. " " L2 // &. '. | 1977. | ,. 22, . 4. | 0. 535{542. 3] ," 3. 4. 0 " ' L2 // &. '. | 1979. | ,. 25, . 2. | 0. 217{223. 4] 6" . "

" L2

// 7 8

. 0 ". | 9

'

"

-

" ",

1986. | 0. 3{10. 5] ;" 4. 4. " L2 " , mr : * , H (g h) = (H (g) + H (h)) ^ (mr ). 4. g 2 Zft. g = (gr )r2I(f )nB Zft , H (g) = 0. g = (gr )r2B g 6= 0, g Zft . . ) , H (g) = 0 ! ! , ! r- - g = (gr )r2I (f )nB " ! . @ gr = a0 + a1 r + : : :+ an;1 nr ;1 | " ! Kr , + ur = b0 + b1r + : : : + bn;1nr ;1, " ur gr = 1 Kr . 1", ! g r Q^ px] ! ! , ! + ^ px], u1 g + v1 r = 1. 1 u1 v1 2 Q ^ px] *+ ; p, Z u g + v r = p : (2) p, uB gB + vB rB = 0 Zpx], % rB * r. 9! rBjuBgB , , rBjuB rBjgB. # ^ px]. 5 " ! g r rBjuB, u = rs + pl, ! r s l 2 Z * " * (2), (rs + pl) g + v r = p ,

284

. .

r(s g + v) + p l g = p . r , r s0 + l g = p ;1, ! ! . ^ px] , * r- , ur gr = 1 Kr , 9! u g + v r = 1 Z g = (gr )r2I (f )nB . @ g = (gr )r2B g = 6 0, g Zft . 5. g h 2 Zft, t = (mr )]. g h , H (g) 6 H (h). . @ g h, h = g u, ! H (h) = H (g u) = = (H (g) + H (u)) ^ (mr ), , , H (g) 6 H (h). ) , H (g) = H (h), H (g) = (lr ) = (kr ) = H (h), kr = lr * r 2 I (f ). r- ! * ! p ! ! : gr = pkr gr0 , hr = plr h0r . 9! hr = plr h0r = pkr h0r = gr (gr0 );1 h0r . ? pkr = gr (gr0 );1 ! gr0 . @ H (g) < H (h), kr < lr * r 2 I (f ). 9! hr = plr h0r = = pkr +sr h0r = pkr h0r psr = gr (gr0 );1 h0r psr . # * ! h * ! g. 5 * 5 ** + *. 1. t- g1 g2 : : : gs 1 ! d, " # H (d) = H (g ) ^ H (g2) ^ : : : ^ ^ H (gs ). 2. t- g1 g2 : : : gs 1 ! v, " # H (v) = H (g ) _ H (g2 ) _ : : : _ _ H (gs ).

1] A. Fomin, O. Mutzbauer. Torsion-free abelian -irredusible groups of nite rank // Comm. Alg. | 1994. | Vol. 22. | P. 3741{3754. 2] . . . ! -"!# $ %& // " $". | 1989. | (. 28, ) 1. |*. 83{104. ' ( 1999 .

. .

514.762

: , , ! " .

# $ %& & ' &( %) ) $! , ! " , ' &( ( )& ( ! " & * $! ! " 4.

Abstract

Yu. F. Pastukhov, Necessary conditions in the inverse variational problem, Fundamentalnaya i prikladnaya matematika, vol. 7 (2001), no. 1, pp. 285{288.

In the paper the inverse variational problem in strati0ed speed spaces of arbitrary order is invariantly formulated, Lagrange cut is de0ned, and an analytic reformulation of the problem is given. We give a necessary condition of a system of ODE of even order not less than 4 to be a Lagrangian system. Tk Xm |

k

Xm , kl : Tl Xm Tk Xm , l > k > 0, |

( k = 0 | |

Xm ). , Xm |

. U (v0 2xn;1) | v0 2xn;1 T2n;1Xm . 1. # $ f : TkXm Tl Xm (0 6 k < l)

!

!

% , &' : f - Tl Xm Tk Xm

Qid Qs

+ l k

Tk Xm 2.

f T2nXm , f = f (U ) = fvx2n 2 T2nXm j vx2n = f (vx2n;1 ) vx2n;1 2 U (v0 2xn;1)g

% ( )

, $

% f : T2n;1Xm U (v0 x2n;1) ! T2nXm : , 2001, & 7, 1 1, . 285{288. c 2001 , ! "# !! $

286

. .

3.

"L = "L (U ) = fvx2n 2 T2nXm j "(x)L(vx2n ) = 0 2 Rmg T2nXm L : n2n;1U ! R | % $

+ , L(x x_ : : : x(n)) |

(x), % $ %

U (v0 2xn;1) T2n;1Xm , , - &, % % $ %

"L T2n Xm . . "(x)L : 22nn;1U ! R | + ,

(x) Xm T2nXm n (n) X "(x) L (x x_ : : : x(2n)) = (;1)k Dkt @L(x@x: :(k: ) x ) i = 1 m: k=0 i

i

0 11],

% %3

(x) Xm

T2nXm . 4 &' 11,2]: f : T2n;1Xm U (v0 2xn;1) ! T2nXm | . 4' & U~ (v0 2xn;1) U (v02xn;1) % $

+ L : n2n;1U~ ! R, f (U~ ) = "L (U~ ). 4.n # + L : Tn Xm ! R % % n $

vx 2 T Xm ,

(x) Xm 2L det @x(n@) @x (x x_ : : : x(n)) 6= 0 (n) L(x x_ : : : x(n)) |

+ L : Tn Xm ! R

(x). 0 11],

% %3

Xm . ( ). f : T2n;1Xm U (v0 2xn;1) ! T2nXm | , U~ (v0 2xn;1) U (v0 2xn;1) | v0 2xn;1 2 2 T2n;1Xm , L : n2n;1U~ ! R |

. f (U~ ) = "L (U~ ) , "(x)L f jU~ (v0 2 ;1 ) : T2n;1Xm U (v0 2xn;1) ! Rm 0 2 Rm

U~ (v0 2xn;1). k

n x

i

287

5. f : T2n;1Xm U (v02xn;12)n;!1 T22nn;2n1;X1m | -

,

v0x 2 T Xm . 4 f

% $ % v0 2xn;1, ' U~ (v0 2xn;1) U (v0 2xn;1) % $

+ L : n2n;1U~ ! R, f (U~ ) = "L (U~ ). 7, +

%8 (

%3 3 ) %3

3 ' % $

% 0 {: $, 8 & 83 %3, $ ;

% : ' & $ % %

%

%3 ++ %3 ? ;% , n > 1 ' % 3 % . . x(2n) = fi(xk0 x_ k1 : : : x(2n;1)k2 ;1 ), i kl = 1 m, | (x) Xm f : T2n;1Xm U (v0 2xn;1) ! T2nXm n > 1 ! U~ (v0 2xn;1) U (v02xn;1) (U~ (v02xn;1) '), ' : U~ ! R2mn | '(U~ (v02xn;1)) = U~(x) (x0 : : : x(20 n;1)) R2mn '(vx2n;1 ) = (xi0 x_ i1 : : : x(2n;1)i2 ;1 ) il = 1 m l = 0 2n ; 1: n X X :::j (x : : : x(n))x(k1) j1 : : : x(k ) j fi (x x_ : : : x2n;1) = Ckij11:::k n

i

n

r=1 n+16k1 :::k r 6rn;1 n r(n+1)6 P ki 6(r+1)n

r r

r

r

i=1

:::jr (x x

Ckij11:::k _ : : : x(n)) | n2n;1(U~(x) ), r 2 n ; 1 2 mn n : R Rm(n+1) | : n2n;1(xk0 x_ k1 : : : x(2n;1)k2n;1 ) = (xk0 x_ k1 : : : x(n)kn ):

!

. .

fi (x : : : x2n;1) = ('0 ij (x : : : x(n)) + '1 ijp(x : : : x(n))x(n+1) p ) x(2n;1)j + + gi(x : : : x(2n;2))

'0 ij (x : : : x(n)) '1 ijp(x : : : x(n)), i j p = 1 m, | n2n;1U~ (x), n2n;1 : R2mn ! Rm(n+1) | , gi(x : : : x(2n;2)) | 22nn;;21U~(x) ,

288 22nn;;21 :

. .

R2mn ! Rm(2n;1) | :

22nn;;21(xk0 : : : x(2n;1)k2 ;1 ) = (xk0 : : : x(2n;2)k2 ;2 ): n

n

1] . . .

. | !. "#$, & 1328-"-96. 2] . -. . / 0012 2. | 3.: 3, 1989.

% ! & 1998 .

- . .

517.929

: - , , ! ", # , $% , % &' .

-" '" # # ' %( # ' ! ".

Abstract L. E. Rossovskii, Strongly elliptic dierence-dierential operators in semibounded cylinder, Fundamentalnaya i prikladnaya matematika, vol. 7 (2001), no. 1, pp. 289{293.

In the paper we consider di/erence-di/erential operator in semibounded cylinder and obtain necessary and su0cient conditions for a G1arding-type inequality using a symbol of the operator.

- X AR = DR D jjjj6m

Q = f(x1 x0) 2 Rn : x1 > 0 x0 2 Gg G ; Rn;1 | P ( @G 2 C 1 n > 3), 1 ; 1 @ n 1 @ D = i @x1 : : : i @xn , R u(x) = aj u(x1 + j x0), aj 2 C . %0 j j j 6 J u(x1 + j x ) = 0 x1 + j 6 0. &' AR () Q* , + u 2 C_ 1 (Q) (1) Re(AR u u)L2(Q) > c1kuk2W m (Q) ; c2kuk2L2 (Q) 2%

34 24" ' # ( , " 5 95{01{00247. , 2001, # 7, 5 1, . 289{293. c 2001 , !" #$ %

290

. .

c1 > 0, c2 u. 0 W n (Q) 1 ) , 23+ L2 (Q) P 3' ) m )4 , kuk2W m (Q) = kD uk2L2(Q) . jj6m 5

- + + 6. 7. 1) ) 81]. : )) + , ) ( , 3 + . ; + ) , . ; , (0 d) G + , d 2= Z. % + ) + ( )

- (. )2 82], 3 + k ! LN2 (>1 ),

(UN u)k (x) = u(x1 + k ; 1 x ) (x 2 >1 @ k = 1 : : : N ): = RN ) N N ( ( (RN )km = am;k jm ; kj 6 J 0 jm ; kj > J: 0

S N

S N

(3)

A R : L2 >k ! L2 >k () k=1 k=1 2 RN )- ) LN2 (>1 ): R = UN;1RN UN : (4)

-

5 , (2) (3) (UN Ru)k = =

X

jj j6J

N X

aj u(x1 + j + k ; 1 x0) =

X jm;kj6J m=1:::N

291

am;k u(x1 + m ; 1 x0) =

(RN )km (UN u)m (x) = (RN UN u)k (x) (x 2 >1@ k = 1 : : : N ):

m=1

1. RN + RN (RN ) N = 1 2 : : :. (R + R ): L2 (Rn) ! L2 (Rn) . . ;' 4 4 ) 4 u 2 L2(Rn0). 5 )+ h 2 R N ) uh , uh (x) = u(x1 + h x ), S N >k . % (4), 2 L2 k=1

Re(Ru u)L2(

Rn = Re(Ruh uh)L )

2

;S N

k=1

k

=

= Re(UN Ruh UN uh )LN2 (1 ) = 21 ((RN + RN )UN uh UN uh )LN2 (1 ) > > ckUN uh k2LN2 (1) = ckuk2 ; SN = ckuk2L2( n) L2

k=1

k

R

c > 0 N = 1 2 : : :. ; L2 (Rn) 2 + ) 4 u 2 L2(Rn). 7 ) . %+ ) 4 C, Re(Ru u)L2( n) = (Re r( )~u u~)L2 ( n) P r( ) = r(1 ) = aj eij1 | . = ,

R

R

jj j6J

2 ' (R + R ): L2 (Rn) ! L2 (Rn) () 2 . =4 1 2. + RN P RNij aj e 1 > 0 (1 2 R). N = 1 2 : : :. Re jj j6J

2. !

. AR ! Q* ,

292

. .

Re

X

X

jjjj=m jj j6J

aj eij + > 0 ( 2 R 0 6= 2 Rn):

. & + . = Qk = Q \ >k

) N

S N 4 ) 4 u 2 C_ 1 Qk . A k=1 V = Un u | )- ) C_ 1N (Q1 ). E (4), X Re(AR u u)L2(Q) = Re (R D u Du) SN =

= Re = Re

k )

(

jjjj6m

X

k=1

jjjj6m

(UN R D u UN D u)LN2 (1 ) =

X

jjjj6m

(RN D V DV )LN2 (1 ) = Re

X

(RN D V DV )LN2 (Q1 ) :

jjjj6m

6 , kukL2(Q) = kV kLN2 (Q1 ) , kukW m (Q) = kV k2W mN (Q1 ) . A) , ) (1) X Re (RN D V D V )LN2 (Q1 ) > c1 kV k2W mN (Q1 ) ; c2 kV k2LN2 (Q1 ) 2

2

2

jjjj6m

) c1 > 0, c2 N V 2 C_ 1N (Q1). % P 4 ( Q* 1

+ D RN D , ) (. jjjj6m

P

81{3]), (RN + RN ) + 0 6= 2 Rn 2 jjjj=m ) , 3 N . ; 2 X Re r ( ) + > 0 ( 2 R 0 6= 2 Rn): jjjj=m

5 . % u 2 C_ 1 (Q). E % <, Re

X

jjjj=m

DR D u u

% 4 X

k1 > 0.

jjjj=m

L2 (Q)

X

=

Rn :

(Re r(1 ) u~ u~)L2 (

jjjj=m

Re r ( ) + > k1jxij2m ( 2 R 2 Rn)

; 2 = 1 . A

X

jjjj=m

Re r(1 ) + u~ u~

R > k kjj u~kL Rn :

L2 ( n)

1

m

2

2(

)

)

-

293

1 % <, )2 () + m + W (Q), Re

X jjjj=m

D R D u u

L2 (Q)

> k1kuk2W m (Q) :

= ) 0, 0 6 < 1, + 1 6 3) kf(t + 9t) ; f(t)k 6 ct; j9tj" " 2 ( ;;1 1], 2 ,0 1) 4) v0 2 D(A ) 2 ( 1],

D: = E , v0 2 D(A ) 2 (minf2 ;;1 g 1],

D: 6= E ( A ,2]). ! (1) " , ! # Zt

v(t) = U(t)v0 + U(t ; s)f(s) ds:

(3)

0

% (1) ,3]. 8 , A (D: = E), = 1 = 0. ; " A ,4]. < + " , j arg j < , j j > R > 0 A * (A + I);1 k(A + I);1 k 6 cj j;r (4) r = 1. = ( = 0, = 1. < " ,5] ,

A * ( > ) (4) ' r 2 (0 1]. < + = 1 ; r, 2 ,0 1), = 1 + , " 2 ( 1], 2 ,0 1 ; ), = 1. < ,6] ' (4) " f : Re > ;c(1 + j Im j)r1 g c > 0 r 6 r1 : @ r = r1 = 1r ; 1, = 2 + 1 (1) , r 2 ( 23 1], 2 ,0 21 ), " 2 (2 1], = 1, = 0.

...

297

8 , ' #'( 1. 8 > , '# "# ' r, ' ' . 8 ,2] ' ' ,

'# + ' ' >7 " ( ,2]

' ' + , < 2). < 1 ' + | > "' ("' , 1 +. < 1 (

' A, v0 2 D(A ) 6 1. = 1 ,3{6]. 2. @ 7 ' 1. 8 (*# .

1. U(t) A( ), A.

" ! ,

# (3). . @ v(t) | (1). 6 ' > v0 (s) + Av(s) = f(s). @ -( U(t ; s), s # x t ; y (0 < x < t ; y < t). 0 '# x y, , 3) ', > ( ,

(5) x y ! +0. @+ *

' (3). C (3) . Rt (3) g(t) = U(t ; s)f(s) ds.

2.

g(t) 2 D

0

# f(t) $ 3) 1.

kAg(t)k 6 ct;; kf k"

kf k" kf kC0" = kf kC0 +

sup

06t 0. 6 ( g(t) > Zt

g(t) = A f(t) ; U(t)A f(0) ; U(t ; s)A;1 f 0 (s) ds: ;1

;1

0

C + g(t) t > 0 (7). @ f(t) ( G7 . # ( t > > 0). J +

299

...

G7 " =" kf k" 6 ckf k"" =" kf k1; C0 0 < "0 < " 6 1. @ + fn (t) ; f(t), ,

> > C0" , ;;1 < "0 < " ( (6)): 0

0

0

0

Zt A U(t ; s),fn (s) ; f(s)] ds 6 ct;; kfn ; f k"0 0

6

" =" 6 ct;;kfn ; f k"" =" kfn ; f k1; C0 : 0

0

@ kfn ; f k" 6 2kf k" , # , Agn(t) ! Ag(t) E t > > 0. @+ gn0 (t) ! f(t) ; Ag(t) ' t > 0. 8( > '. v1 (t) = U(t)v0 (3). H v1 (t)

(* (1). < ,2] , + ' t > 0, v0 2 D(A ), 4) ' 1. @+ v(t) (3) (1). = ' 1. 3. 6 1 - ' ' ' . @v + (;1)m @ 2m v = f(t x) t 2 (0 1] x 2 ,0 1] (8) @t @x2m v(n ) (0) + v(n ) (1) + T v = 0 = 1 2 : : : 2m ; r (9) Z1 0

'k (x)v(x) dx = 0 k = 1 2 : : : r

(10)

v(0 x) = v0 (x) x 2 ,0 1]: (11) % | ' , j j + j j 6= 0, 0 6 n1 6 : : : 6 n 6 6 n +1 6 : : : 6 2m ; 1, T | ' ' ' Wpk ;1(0 1) p 2 ,1 1), = 1 2 : : : 2m ; r, 0 6 r 6 2m, f'k (x)g | ''# '# ,0 1]. 8 , (9){(10) ( ' ,5]. K( " 2m Lp (0 1). 2 ; 2e 2 : ) ! X = Rn "" # : ""

, ( (

Ancient Narrative 1. (2000-2001)

Ancient Narrative 1. (2000-2001)

Επιστημονική Επετηρίδα Βελλάς 1(2001)

Επιστημονική Επετηρίδα Βελλάς 1(2001)

2001

2001)

2001

2001)

2001)

2001

2001

Algorithms in Bioinformatics, 1 conf., WABI 2001

Algorithms in Bioinformatics, 1 conf., WABI 2001

Дезинформационный листок украинской фантастики (1 апреля 2001)

Дезинформационный листок украинской фантастики (1 апреля 2001)

Газета День Литературы # 52 (2001 1)

Газета День Литературы # 52 (2001 1)

Космическая одиссея 1 (Комическая одиссея 2001 года)

Космическая одиссея 1 (Комическая одиссея 2001 года)

Газета День Литературы # 52 (2001 1)

Газета День Литературы # 52 (2001 1)

Информационный листок '22-68', N 1 (1), февраль 2001 года

Информационный листок '22-68', N 1 (1), февраль 2001 года

2001 (109)

2001 (87)

2001 (105)

Ecscw 2001

2001 (95)

Роскон-2001

2001 (96)

2001 (92)

Re 2001

Фантастика 2001

Фантастика 2001

Веребушка - 2001

Веребушка - 2001

2001 (82)

2001 (94)

2001 (88)

2001 (90)

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